01/01/10

EXPANDED CHESS

FOR THE NEW MILLENNIUM

Taking Traditional Chess Into Many New

Dimensions

WEBSITE MANUAL

5

By

Daniel Abuda

© 2009, 2010 Copyright

Daniel Abuda

All Rights Reserved.

1

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WEBSITE MANUAL

Version 1.0

This manual explains Expanded Chess in detail and complements the Expanded Chess website

http://expandedchess.com.

Table of Contents

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Chapter 1 – Introduction ____________________________________________________ 1

Origins _______________________________________________________________________ 1

General Themes to Expand Traditional Chess _______________________________________ 2

Variants of Chess _______________________________________________________________ 5

Chapter 2 – Classification and Analysis of Traditional Chess Pieces _________________ 7

Classification of Traditional Chess Pieces ___________________________________________ 8 Descriptive Classification _______________________________________________________________ 8 Movement Classification ________________________________________________________________ 9

Value of Traditional Pieces ______________________________________________________ 13

Analysis of Traditional Chess Movements _________________________________________ 15

Chapter 3 – Addition of Expanded Chess Pieces _______________________________ 18

The New Pieces _______________________________________________________________ 18

Description of Expanded Chess Pieces ____________________________________________ 21

Chapter 4 – Movements of Expanded Chess Pieces_____________________________ 22

Expansions of L-Type Movements ________________________________________________ 22

Combining Components of Movements and Pieces __________________________________ 25

Chapter 5 – Values Of Expanded Chess Pieces ________________________________ 29

Methods of Establishing Values for Chess Pieces ____________________________________ 29

Approach for Establishing Values for Expanded Chess Pieces _________________________ 29

Creating a Value System for Expanded Chess Pieces ________________________________ 31 Start of Game Configuration ____________________________________________________________ 31 Types/Components of Movements Summary Matrix – Appendix D ______________________________ 32 Types/Components of Movements Summary Table – Appendix E _______________________________ 33

Weighting/Rating Summary Matrix – Appendix F __________________________________ 34

Value of Expanded Chess Pieces _________________________________________________ 34

Chapter 6 – Size of Game Board ____________________________________________ 49

Minimum Board Size __________________________________________________________ 49 Protective Pawns _____________________________________________________________________ 49 Four Rows Deep _____________________________________________________________________ 50 Five Rows Deep _____________________________________________________________________ 50 Six Rows Deep ______________________________________________________________________ 51

Table of Contents

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Minimum Board Width ________________________________________________________________ 52

Maximum Size of Board ________________________________________________________ 53

Chapter 7 – Arrangement of Game Board ____________________________________ 54

Shape of the Game Board _______________________________________________________ 54 Square and Rectangle Shapes ___________________________________________________________ 54 Board Arrangement and Notation ________________________________________________________ 55 Other Shapes ________________________________________________________________________ 57

Number of Spatial Dimensions of the Board ________________________________________ 59

Number of Colors Used on the Board _____________________________________________ 60 Three Color Board ____________________________________________________________________ 60 Four Color Board _____________________________________________________________________ 63

Chapter 8 – Number of Players During Game _________________________________ 67

Traditional Chess _____________________________________________________________ 67

Expanded Chess ______________________________________________________________ 67

Chapter 9 – Types and Number of Pieces at Start of Game _______________________ 72

Open Rows ___________________________________________________________________ 72

Defining Low, Intermediate and High Level pieces __________________________________ 72

Scaling Up ___________________________________________________________________ 76 Scaling Up Types of Pieces _____________________________________________________________ 76 Scaling Up Number of Pieces ___________________________________________________________ 77 Small Board - Table 9-3A ______________________________________________________________ 79 Medium Board – Table 9-3B ____________________________________________________________ 80 Large Board – Table 9-3C ______________________________________________________________ 81

Relationship to Major and Minor Pieces in Traditional Chess _________________________ 81

Chapter 10 – Placement of Pieces at Start of Game _____________________________ 84

Guidelines ____________________________________________________________________ 84

Low Level Pieces ______________________________________________________________ 85

Intermediate Level Pieces _______________________________________________________ 85

High Level Pieces ______________________________________________________________ 85

Checkmate Level Pieces ________________________________________________________ 86

Positions of Level of Pieces @ Start of Game _______________________________________ 87

Chapter 11 – Penalties and Rewards for Capture and Check _____________________ 96

Table of Contents

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Capture of Playing Pieces _______________________________________________________ 96

Penalties and Rewards for Capture _______________________________________________ 97

Penalties and Rewards for Check ________________________________________________ 99

Penalties and Rewards Summary _______________________________________________ 101

Chapter 12 – Promotion of Playing Pieces ___________________________________ 102

Pieces That Can Be Promoted __________________________________________________ 102

Range and Value of Promotable Pieces ___________________________________________ 102 Range of a Promotable Piece ___________________________________________________________ 103 Value of a Promotable Piece ___________________________________________________________ 103

Quantifying Promotability _____________________________________________________ 104 Effect of Strength (value) of Piece Being Promoted _________________________________________ 105 No. of Pieces of Specific Type at Start of Game ____________________________________________ 105 Size of the Game Board _______________________________________________________________ 105

Calculating Promotion Reward Factor ___________________________________________ 106

Combining the Two Approaches for Promotion ___________________________________ 109

Chapter 13 – Special Rules for Expanded Chess ______________________________ 111

Castling _____________________________________________________________________ 111

En Passant __________________________________________________________________ 112

Replacement of a King on a Larger Board Game __________________________________ 114

Multiple Players / Multiple Movements per Turn __________________________________ 115

Slide Bishop _________________________________________________________________ 115

Chapter 14 – Describing and Cataloging New Games__________________________ 117

Major Game Parameters ______________________________________________________ 117

Chapter 15 – Conclusions and New Expanded Chess Games ____________________ 122

New Pieces __________________________________________________________________ 122

New Boards _________________________________________________________________ 123

New Rules ___________________________________________________________________ 123

Final Challenge – The New Expanded Chess Games ________________________________ 124

Bibliography ____________________________________________________________ 125

Table of Contents

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Appendix A – DESCRIPTION OF PIECES & MOVEMENT/CAPTURE DIAGRAM Appendix B – DESCRIPTION OF PIECES ON CLASS BASIS Appendix C – GAME DESCRIPTION & GAME ARRANGEMENT FOR NEW GAMES

Appendix D – TYPES/COMPONENTS OF MOVEMENTS SUMMARY MATRIX.

Appendix E – TYPES/COMPONENTS OF MOVEMENTS SUMMARY TABLE

Appendix F – WEIGHTING/RATING SUMMARY MATRIX

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Chapter 1 – Introduction

Origins

Expanded Chess originates from the most significant historical event in chess in our lifetime. This event occurred in 1997 when, for the first time, a computer defeated the best chess player in the world. The IBM supercomputer Deep Blue defeated Garry Kasparov in a six game match with two wins, one loss and three draws in May, 1997 in New York City, USA. Until 1997 there was still a popular perception that a computer could not defeat the best human player in the world. Since that match, the software has become much better and the hardware much less expensive and now no human can defeat the best chess playing computer program in tournament play.

I feel that chess now requires an industrial strength, rigorous intellectual upgrade to meet modern technology head-on and survive in a world of the modern computer and the Internet. The traditional game now risks becoming stale and intellectually unappealing because computers can now win against the best humans and computers now play the best chess. Chess needs its complexity increased to the level where humans can again easily defeat the best computer to meet the challenges of the computerized world and its honor restored. Expanded Chess is the method to restore this honor by creating a series of Expanded Chess games which are new, undocumented, unanalyzed, untrended, and therefore ripe for exploration and discovery, which automatically ensures that they have not been over-analyzed and cataloged for the past 100 years.

Increasing the complexity of chess also obligates the technology to develop more elegant theories, algorithms and software to win against humans with this new series of games in play. Relying on historical experience and brute force number crunching will be less successful for

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the programmers and humans will occupy the high ground in this battle of wits.

General Themes to Expand Traditional Chess

The purpose of this manual is to expand traditional chess into a rich, new set of games that will carry chess into the new millennium and beyond. Expanded Chess accepts the challenge and presents a rigorous analysis and thoughtful expansion of traditional chess in a way that is intuitive and follows directly from the traditional game. This approach has many benefits. The most obvious benefit is that you do not lose the huge base of traditional chess players. This is accomplished by expanding from traditional chess and following the same philosophies, but expanding the approach of those philosophies in ways that are logical, meaningful and intuitive to traditional chess player. Our goal here is to retain our flagship of chess with its rich history and tradition but strengthen and expand the game to meet the demands of the technological future and better understand and appreciate the role games as an important human endeavor.

Expanded Chess proposes to develop a larger base of standard pieces with standard moves. There is a major benefit to this approach because many versions of the game can be developed with the same piece base. Players can move intuitively from one expanded game to another because they already know any new pieces and their moves.

This manual is really a guide for developing new versions of Expanded Chess. It begins with traditional pieces with standard moves and recommends the following:

addition of Expanded Chess pieces

movements of Expanded Chess pieces

minimum and maximum size of the game board

value of Expanded Chess pieces as the size of the game board increases

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arrangement of the game board as the size of the game board increases

number of players as the size of the game board increases

number of moves per player as the size of the game board increases

type and number of pieces at the start of the game as the size of the game board increases

placement of the pieces at the start of the game as the size of the game board increases

rules for the capture of pieces

rules for Checkmate

rules for promotion of pieces

other rules for new games

recommendations for describing and cataloging expanded versions of chess

and inclusion of eight new Expanded Chess games

Before we begin to expand traditional chess, we need to break down the game down into its most salient characteristics, analyze the structure of these characteristics and then expand the characteristics in multi-dimensional ways.

The analysis in this manual uses a three-part approach. The first part of this approach classifies the most salient features of a major aspect of traditional chess. The second part takes these most salient characteristics and expands them in multiple dimensions. Here we study the implications and limitations of our expansion process. The third part then reviews the expansions of the second part and synthesizes them into coherent generalized format. Finally, in the third part of the approach, methods of cataloging the expansions are presented in order to create each specific expanded version. This

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approach is used throughout the manual including each major aspect of the game that is expanded.

The general approach for expanding the traditional realm of chess is to keep changes as intuitive as possible. This is a subjective exercise. However, I want to be as non-arbitrary as possible and try to explain each expansion leap in terms of people‟s present general knowledge of traditional chess and common sense.

Let us examine the traditional game of chess and extract the implicit assumptions and procedures. Then we will state each assumption or procedure explicitly and expand it. Finally we will test the expanded assumption or procedure and determine if the intent of the original assumption or procedure can still be met in the expanded format. If this is the case, the expansion can be permutated to some practical limit and adopted.

There are several intuitive and common sense themes that make up traditional chess and are implicit when one expands chess into new dimensions. The traditional chess pieces should retain their traditional properties to the greatest extent possible in any expanded version of traditional chess. If a traditional property is altered, then the alteration should be done in a systematic way, with the least significant property being changed first. Themes that are examined include:

When the size of the game board becomes either larger than traditional chess (8x8 squares) then the number of pieces should increase in a relatively proportional manner.

New pieces that are added should be added in such a way that they have some connection with the traditional chess pieces.

Completing the game by capturing the “King” can be expanded to capturing other than the “King” to end the game.

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Making one move then giving the opponent one move can be expanded so that a player may have, under certain circumstances, more than one move at a time.

The number of players playing the game at one time can be expanded. More than one player per side can participate in the game and increase the number of moves per side for each turn.

The arrangement of the strength of the pieces on the board should generally remain strongest to weakest from middle to outside and from back to front.

There needs to be a front line piece (Pawn type) that allows the game to start in an orderly manner. The Pawn-like piece has key attributes that requires preserving and include:

o Limited range on each move

o Forward only direction of movement to make the start of the game irreversible.

o Each piece should be protected by other pieces in the front row so they can advance without being subject to immediate capture.

Pieces other than Pawns can be promoted.

All these aspects of traditional chess will be examined in extensive detail in the forthcoming chapters and chess will be expanded after thorough exploration of the major aspects.

Variants of Chess

A common approach to develop a new variant of chess has been to add a new piece or two with new movements and perhaps expand the board by a row or column or two. Another approach is to modify existing variants even more. Some radical variants use completely new boards or new types of movements and have significantly modified rules. There are literally hundreds of variants of chess and over one hundred new

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pieces based on these variants. The major issue that I have with most of these variants of chess is that you cannot use what you learned in one variant of chess to play another variant of chess. You have to keep learning over and over new rules, new moves and new pieces each time you learned a new variant of chess. This lack of standardization is the most significant issue impeding the growth for any variant of chess, which in turn results in the following drawbacks:

The possibility of a new piece with a particular name and image having more than one movement pattern.

Possibility of differing and contradictory rules for each variant

Possibility of non-intuitive structural modifications

This non-standardized approach is, in my opinion, anarchy. It results in a chaotic hodge-podge of pieces, rules, and game boards with most variants having few followers and a short lifespan. Most of these variants never obtain any kind of critical mass and most traditional chess players do not seriously consider playing variants. My approach is to avoid all these drawbacks and to use traditional chess as the foundation upon which to build Expanded Chess games in an intuitive, structured, logical manner so that people will innately feel that nothing is lost and no major change of direction is needed when playing an Expanded Chess game. The progression will be natural and seamless.

With this expansion plan in place, we can now begin to describe, classify and analyze traditional chess so we can develop detailed dimensions to begin the expansion of traditional chess. This analysis and classification is the subject of the next chapter.

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Chapter 2 – Classification and Analysis of Traditional Chess Pieces

To begin this chapter some basic definitions are needed. In traditional chess, a piece is often considered all the non-Pawn material on the chess board. As well, there are minor pieces (Knights and Bishops) and major pieces (Rooks and Queens) in traditional chess. Major pieces are defined as pieces that in combination with a King can checkmate the opposing King. In this manual, we will not use these definitions for Expanded Chess because they end up being too constrictive or are currently undefined. In Expanded Chess, pieces are any and all material on the game board. The commonly used terms of rank and file are also not used in Expanded Chess because they are not immediately obvious to the inexperienced chess players. The more common terms of row and column are used instead of rank and file for immediate recognition and understanding, since the game board arrangement is constantly being discussed and it is important for the reader not to get bogged down always trying to recall that a rank is a horizontal lines of squares and a file is a vertical line of squares on the game board. Also note that the term Rook is used for traditional chess and Castle is used for Expanded Chess. The different name is used to recognize that the role of the Rook is modified for Expanded Chess.

The classification and analysis of traditional chess pieces is necessary to be able to understand the inherent underlying structure of each dimension of chess so that expansion can take place along that dimension. Once these underlying structures are identified and understood, each one can be explored for potential expansion. To the greatest extent possible, expansion will take place, and in a structured manner so it is relatable to traditional chess. The classification and analysis of traditional chess pieces is divided up into the following categories:

Classification of traditional pieces including:

o Descriptive Classification

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o Movement Classification

o Valuation of traditional chess pieces

o Analysis of traditional chess pieces

This classification, valuation and analysis form the fundamental framework for the expansion of the number of chess pieces.

Classification of Traditional Chess Pieces

Descriptive Classification

When applying a descriptive classification to traditional chess, an obvious approach used is to base the classification of the pieces on the real life roles of the pieces. At present, there are six (6) pieces in traditional chess. These are the Queen, King, Castles, Bishops, Knights, and Pawns. Each piece, except the Castle is human. The human pieces are part of a medieval European society. With consideration for expansion to include new pieces, traditional chess pieces are classified as presented in Table 2-1.

Table 2-1 Descriptive Classification of Traditional Chess Pieces

No. Classification Pieces

1. Ruling 3 Queen 5 King

2. Architectural 9 Castle -----

3. Religious f Bishop -----

4. Military n Knight -----

5. Special z Pawn -----

The Queen and the King are obvious candidates for the Ruling Classification. The Castle is a building and is placed in an Architectural Classification. The Bishop is a member of the Church and is part of the

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Religious Classification. As for the Knight, it is an instrument of war and belongs in the Military Classification.

Although a Pawn is usually described as a foot soldier, and in principle could be placed in the Military Classification, a more metaphorical chess-based definition of the Pawn as a small, weak, sacrificial piece which is easy to manipulate and has special movements is used by Expanded Chess. A Pawn is defined as a Special Character since it has the special ability to be promoted but is not allowed to move in the backward direction, cannot capture directly ahead and cannot move diagonally ahead.

The purpose of the descriptive classification of the pieces is to provide a framework to add new pieces. It is one of the most intuitive ways to expand traditional chess.

Movement Classification

At present, there are six (6) pieces in traditional chess. These are the Queen, King, Castles, Bishops, Knights, and Pawns. Along with these six pieces, there are three (3) types of movement. The movements are orthogonal, diagonal and L-Type movements. The orthogonal and diagonal movements can be further classified as unlimited or limited.

Orthogonal movements advance along only along a single row or single column of the game board in one move.

Diagonal movements advance an equal number of squares along a row and along a column simultaneously and remain on the same color square.

L-Type movements are made by moving X squares along a row and Y squares along a column from a square of origin or vice versa such that :

X does not equal Y

and the sum of X + Y is greater or equal to three.

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This is symbolically expressed as:

X ≠ Y and ∑ X + Y ≥ 3 [Eqn. 2.1]

The three types of movements can be combined together in various ways to create a basis for the movements for new pieces in Expanded Chess. These types of movements are presented in Tables 2-2, 2-3 and 2-4 following.

Table 2-2 8x8 Game Board Orthogonal Moves

08 ┼┼┼

07 ┼┼┼

06 ┼┼┼

05 ┼┼┼

04 ┼┼┼ ┼┼┼ ┼┼┼ ●●● ┼┼┼ ┼┼┼ ┼┼┼ ┼┼┼

03 ┼┼┼

02 ┼┼┼

01 ┼┼┼

a b c d e f g h

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Table 2-3 8x8 Game Board Diagonal Moves

08 XXX

07 XXX XXX

06 XXX XXX

05 XXX XXX

04 ●●●

03 XXX XXX

02 XXX XXX

01 XXX XXX

a b c d e f g h

Table 2-4 8x8 Game Board L-Type Moves

08

07

06 LLL LLL

05 L L

04 ●●●

03 L L

02 LLL LLL

01

a b c d e f g h

With consideration for expansion to include new pieces, traditional chess piece movements are classified as follows in Table 2-5, which places the movements shown in Tables 2-2 to 2-4 with each of the traditional chess pieces. Table 2-5 describes each piece‟s movements as

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orthogonal, diagonal or L-Type or the appropriate combination of these and also shows whether each type of movement is limited, unlimited or special. The purpose of movement classification of the pieces is to provide a framework to add new pieces to Expanded Chess. It is another intuitive way to expand traditional chess.

Table 2-5 Movement Classification of Traditional Chess Pieces

No. Piece Type of

Movement Movement Classification

1. 3 Queen

9 Castle

+

f Bishop

=

3 Queen

Unlimited Orthogonal +

Unlimited Diagonal

2. 9 Castle 9 Castle Unlimited Orthogonal

3. f Bishop f Bishop Unlimited Diagonal

4. n Knight n Knight L-Type (2,1)

5. 5 King 3 Queen

Limited Orthogonal +

Limited Diagonal

6. z Pawn

z Pawn

(Special)

Limited Orthogonal (Forward only, optional two

moves for 1st move and then one move

+ Limited Diagonal

(Forward only, capture only, one move)

Promotion

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Value of Traditional Pieces

Traditional Chess typically assigns the following values to pieces as shown in Table 2-6.

Table 2-6 Value of Traditional Chess Pieces

Piece Value

3Queen 9

5King *

9Castle 5

fBishop 3

nKnight 3

zPawn 1

* Rating the King is not considered viable because the King cannot be traded. However, the King has attacking capability and can assist in a checkmate so the piece does have limited value.

This value system is empirical, based on experience and is not based on a specific mathematical formulation or algorithm. As an example of the empirical nature of this value system, a Castle and a Bishop combine together to equal the movements of a Queen; however, the sum of the value of a Castle (5) + a Bishop (3) does not equal the value of a Queen (9). The value system is simple because each piece has a whole number value with the Pawn as unit value. With this approach, it is simple to assess the relative strength of any pieces involved in a trade. For example, one Castle is equal to one Bishop and two Pawns.

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The values listed are also average values over the entire span of the game. Depending on the position of the piece, the number of other pieces on the board and the how far the game has progresses, the value of each piece will vary. Bishops and Knights tend to be of greater value early in the game while Castles, Queens and Pawns tend to be of greater value later in the game when the board is more open and the opportunity for Pawn promotion is most likely.

When new pieces are introduced to the game, it is important that each piece is assigned a value so that players know the relative strength of each piece and can position pieces safely and trade wisely. This can be done based on experience, but it would take years for new pieces to acquire a widely agreed upon value. The problem becomes more daunting if one considers increasing the size of the game board beyond the traditional 8x8 size. Do pieces have different relative strengths on different sized boards? Again, it would take years to answer the question. Empirical values require the experience of thousands of players playing thousands of games that require documentation and analysis.

With 36 pieces in Expanded Chess as shown in Table 3-1, there needs be a quantitative value for each piece and a subsequent ranking based on the value.

The value of a Chess piece is strongly related to its movement capability. Classifying this movement is the first step to developing a systematic method for establishing the value of a chess piece. Once this movement classification is established, each of the components can be evaluated for its contribution to the overall value of the piece. Once the components are evaluated, they can be reassembled in new combinations for new pieces and the summed to determine the value of a new piece. This is the approach that is used to determine the value new chess pieces in Chapter 5.

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Analysis of Traditional Chess Movements

There are three major types of movements in traditional Chess. The movements are orthogonal, diagonal and L-Type movements. These movements are illustrated in Figure 2.1

Orthogonal moves can be limited or unlimited and forward, backward, left or right (sideward). Limited Orthogonal Moves [LOM] can be defined as:

LOM ≤ (n-1)/2 [Eqn. 2.2]

where n is the number of rows or columns on a game board (assuming a square board).

Table 2.-7 presents the classification of orthogonal moves for the traditional chess pieces.

Table 2-7 Components of Orthogonal Moves – Traditional Chess

Piece Component

Forward / Backward Left / Right (Sideward)

Unlimited Limited

0 1 2

3Queen F/B/L/R

5King F/B/L/R

9Castle F/B/L/R

fBishop X

nKnight X

zPawn F F*

*Opening move only

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Diagonal moves can be limited or unlimited and forward or backward. Limited diagonal moves [LDM] can be defined as:

LDM ≤ (n-1)/2 [Eqn. 2.3]

where n is the number of rows or columns on a game board (assuming a square board).

Table 2-8 presents the classification of diagonal moves for the traditional chess pieces.

Table 2-8 Components of Diagonal Moves – Traditional Chess

Piece Component

Forward / Backward

Unlimited Limited

0 1 2

3Queen F/B

5King F/B

9Castle X

fBishop F/B

nKnight X

zPawn F*

*Capturing opponent only

L-Type moves are limited by definition and are forward, backward, left or right. Table 2-9 presents the classification of L-Type moves for the traditional chess pieces.

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Table 2-9 Components of L-Type Moves – Traditional Chess

Piece Component

Forward / Backward Left / Right

Unlimited Limited

0 2,1

3Queen X

5King X

9Castle X

fBishop X

nKnight F/B/L/R

zPawn X

The classification and analysis of traditional chess pieces has been divided up into the following categories:

Classification of traditional pieces including:

o Descriptive Classification

o Movement Classification

Valuation of traditional chess pieces

Analysis of traditional chess pieces

Tables 2-7, 2-8 and 2-9 have provided an analytical basis for determining the structure of the components of the orthogonal, diagonal and L-Type moves for traditional chess pieces. New chess pieces can be created and provided with movements based on expanding and/or recombining these components in a manner the follows rationally from the existing movements. This classification, valuation and analysis form the fundamental framework for the expansion of the number of chess pieces. Adding these new pieces is the subject of the next chapter.

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Chapter 3 – Addition of Expanded Chess Pieces

The New Pieces

Expanded Chess will increase the number of type of chess pieces to thirty-six (36). The new pieces are based on classifications that will extend naturally from traditional chess. At first this might seem like a large increase in the number of types of pieces for Expanded Chess. However, not only are the number of pieces increasing, but the size of the game board is also increasing in size. With the larger game board, new pieces are needed. It is not the intent of Expanded Chess to use all pieces in anyone game. The intent of Expanded Chess is to create a new set of games where a unique subset of the 36 pieces is used for each new game. These pieces will have the same movements in every game that they are used. There are six (6) classifications with six (6) types of pieces in each classification. The results are as shown in Table 3-1.

Table 3-1 Additional Types of Expanded Chess Pieces

Classification

“Y” Index

“X” Index

Type of Movement

6 5 4 Type of

Movement 3 2 1

Ruling

6

Queen/ Knight

0 Empress

2 Chancellor

1 *Emperor

Queen

3

Queen

4

Duke

5

King

Architectural

5

Castle / Knight

6

Fortress

7 Citadel

8

Cathedral

Castle

9

Castle

a Armory

b

Abbey

Religious

4

Bishop / Knight

c

Pope

d

Archbishop

e

Cardinal

Bishop

f

Bishop

g

Priest

h Nun

Military

3

Knight / Knight

Ii Marshall

j Captain

k

Sergeant

Knight

l

Guard

m

Archer

n

Knight

Weaponry

2

Pawn / Knight

o

Catapult

p

Ram

q Mace

Partial Knight

r Lance

s

Hatchet

t

Sword

Special Characters

1

Special u

Angel

v

Saint

w

Wizard

Pawn

x

Squire

y

Knave

z Pawn

*Emperor does not have Knight- type of movement. An overly mobile Emperor is very difficult to checkmate.

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The classification system does pre-suppose that any new pieces will fit into a model that approximates cultural coherence. The chosen model for the classification is a medieval Catholic society. This model is used in Expanded Chess because it is the one that is already assumed in traditional chess. It is also the one that is rich enough in history and literature to support any an expansion, such as the one that is being proposed, with relative ease. The model is also a familiar one to much of the world even though it carries a Euro-centric cultural bias.

Each row lists a Type of Movement which is representative for that class of piece. For example, the Ruling Class Type of Movement is listed as Queen/Knight. This means that each piece in the Ruling class will have movement on the chess board that is a combination of Queen-like and Knight-like moves. The salient features of Queen-like moves or Knight-like moves or any types of movements are explained in detail in Table 2-2.

A combination movement is defined as a piece having the movements of two lesser pieces. In traditional chess, the main example of this concept is the Queen. She has the combined moves of a Castle and a Bishop, which creates the more powerful piece. A significant benefit to using combined moves is the economy of recall required by players. The less arbitrary and random the moves of a more powerful piece are (and the more these moves are a combination of the already learned moves of lesser pieces) then the easier it is for a player to learn and recall the moves of more powerful pieces. As the number of types of pieces is expanded, this concept becomes important because it is difficult recall a large number of types of pieces with no obvious pattern to their movements.

Table 3-1 presents the structure of the pattern of movements. As you move down Table 3-1 in class from Row 6 to Row 1, the Type of Movement becomes a less powerful combination. As you move across Table 3-1 in the same class from Row 6 to Row 4, the Type of Move generally remains the same combination but the movements from unlimited to limited and decrease in range. As you move across Table 3-1 in the same class from Row 3 to Row 1, the Type of Movement

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eliminates the combination but retains the more powerful component of the combination and again the movements from unlimited to limited and decrease in range. Table 3-1 gives a description of each class of pieces and then lists the pieces in that row in order of relative strength. The row and column index generally gives the overall relative strength of the piece. For example, the piece with the 6 6 index is the strongest, 3 3 is a medium strength piece and the 1 1 is the weakest piece. The present number of types of pieces in traditional chess is six (6). An expansion to thirty-six (36) types of pieces does provide a tremendous wealth of possibilities for all the new Expanded Chess games. The main objective of having all these new types of pieces is to standardize, now and for the future, the concept that each type of piece will have the same movements and rules governing its play for all the new Expanded Chess games.

The traditional chess pieces will, keep the same movements in Expanded Chess as they now have in traditional chess. The only noteworthy exception is the slide bishop. The slide bishop will be explained in detail in Chapter 13.

The new pieces selected are ones that fit with the concept that all items in a row in Table 3-1 belong in that class and are also obviously arranged from pieces of greatest importance to least importance from left to right within a row in Table 3-1. Some of the pieces selected are similar to ones that appear in variant games of chess. This includes pieces like the Marshall, Cardinal, Archbishop, Duke, Empress, Chancellor, Sergeant, Guard and Wizard.

Since there are hundreds of variants of chess, this naming situation is unavoidable because the use of common items from medieval culture in variant chess has existed since the origin of traditional chess and is inherently obvious for adding new pieces. The Queen and the Bishop originated as new pieces in a variant of Arab game Shatranj created in fifteenth century Italy, which is now traditional chess. The firz and the fil (elephant), were transformed to the Queen and the Bishop respectively.

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This, however, was not a rigorous survey. It is a subjective selection. The types of pieces selected are pieces that are intuitive extensions of pieces that are now in existence. For example, a Duke would be an obvious choice for a piece that would be related to the King and Queen.

Expanded Chess provide a complete new iconography where traditional chess pieces maintain their standard iconography and each new type of piece has its own new and unique iconography that also includes a three letter abbreviation of the piece to simplify identification and minimize effort to recall the name of each piece. There is no attempt in Expanded Chess to combine the images of two existing pieces to synthesize the image of a new piece if the movements of the two lesser pieces are combined to form a greater piece. This is not done in traditional chess when the movement of a Bishop and a Castle are combined to create a Queen. Expanded Chess maintains this approach and gives each piece a unique identity as is done in traditional chess.

No attempt has been made to harmonize the existing pieces in different variants with each other or Expanded Chess. The problems associated with such an undertaking are manifest. You would need to harmonize pieces, names, movements and iconography for hundreds of variants. What criteria would be used to select one game‟s pieces, names, movements and iconography over any other game? It would be a major challenge and difficult to imagine creating such a set of rules to achieve this harmonization of the existing variants because these variants never followed any expansion guideline when they were created. Expanded Chess will build these entities from the ground up with a structured set of guidelines for expansion and create a coherent set of pieces, names, movement and iconography for all the variants of Expanded Chess.

Description of Expanded Chess Pieces

Appendix A presents a detailed description of each piece including the image of each piece and its movements/capture diagram.

The description of the new types of pieces is provided on a class-by-class basis in Appendix B.

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Chapter 4 – Movements of Expanded Chess Pieces

Expansions of L-Type Movements

From Tables 2-7, 2-8 and 2-9 already presented in Chapter 2, it is evident that these movement tables are sparsely populated tables with many possible component combinations of moves that are still available for use in Expanded Chess. It can also be observed from the tables that the components can be expanded. Limited moves do not need to be fixed at 1 or 2 squares. There is also potential to expand the L-Type movements. Now consider the possible combinations of L-Type or Knight-like moves in Table 4-1.

Table 4-1 Sub-components of Possible L-Type Moves

Moves Left or Right

x=1 2 3 4

Moves Forward or Backward

y=1 N/A A (2,1) B (3,1) C (4,1)

2 A (1,2) N/A D (3,2) E (4,2)

3 B (1,3) D (2,3) N/A F (4,3)

4 C (1,4) E (2,4) F (3,4) N/A

N/A = Not Applicable

Table 4-1 presents the sub-components of possible L-Type moves where the letters A, B, C, D, E and F represent both combinations of pairs of coordinates in the table. The letter “A” represents the moves of the Knight in traditional Chess. For another example, consider the letter “D” representing (2,3) and (3,2) which, as with all the lettered coordinate pairs, are reflections over the main diagonal. The main diagonal is the line of cells in Table 4-1 from top-left to bottom-right filled with “N/A”.

Figure 4.1 shows how all the sub-components from Table 4-1 of an L-Type move are positioned on a game board. Rotating in counterclockwise direction, all eight positions for “D” are:

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Square in Figure 4.1 a 3,2 Right / Forward

b 2,3 Right / Forward c 2,3 Left / Forward d 3,2 Left / Forward e 3,2 Left / Backward f 2,3 Left / Backward g 2,3 Right / Backward h 3,2 Right / Backward

3,2 L-Type Move a= Right /Forward

a,b,c,d,e,f,g,h locations shown

08

07 ccc bbb

06 ddd aaa 2

05 1

04 ●●●

03 1 2 3

02 eee hhh

01 fff ggg

a b c d e f g h

Figure 4-1 3, 2 L-Type Move

Figure 4.-1 illustrates that an L-Type move will cover a maximum of eight positions on a game board as shown by the positions a, b, c, d, e, f, g, and h.

From Table 4-1, x or y can be any number from 2 to:

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xmax = (n -2) / 2 if n is even and [Eqn 4-1a] xmax = (n-1) / 2 if n is odd [Eqn 4-1b]

where n is the maximum number of rows or columns on the board (assuming a square board). In Table 4-1, x and y are limited to 4 because it has been realized that a traditional 8x8 Chess board is not able to accommodate all the positions for a piece that has 4,Y L-Type moves. Since one of the objectives of Expanded Chess is to maintain compatibility with traditional Chess wherever possible and practical, creating pieces with L-Type moves with a maximum range of 4 or greater moves is deemed unnecessary.

Finally, if you want to determine the maximum number U of unique L-Type moves for a square board of n x n rows by columns there are:

U = [xmax 2 - xmax] / 2 unique “L-Type” moves, [Eqn. 4.2]

for a piece that can move in any L-Type move where x does not equal y and x + y is greater than equal to 3.

For an 8x 8 game board, xmax = 3

U = [32 – 3] / 2 = 3.

And for a 16x16 game board, xmax = 7

U = [72 - 7] / 2 = 21.

As shown in Table 4-2, for Expanded Chess, the standard nomenclature for an L-Type move is:

x,y L-Type

where x is greater than y and the sum of x + y is greater than or equal to 3.

Table 4-2 gives a tabular representation of Eqn. 4.2 with the resulting combinations of L-Type moves possible for each square board size. Since we are assuming any piece in Expanded Chess could be used in

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traditional chess, L-Type moves greater than 3,1 or 3,2 are not being used.

Table 4-2 L-Type Moves used In Expanded Chess

Maximum Moves

Forward Backward

Left Right Xmax

Possible Maximum

Unique L-Type Moves

Possible L-Type Moves X , Y

U L-Type Moves from

Eqn. 4-2

Used in Expanded Chess

1 0

2 1 2,1 1 Yes, this is the movement pattern

of the Knight.

3 2 3,1 3,2 3 Yes, these are the movement

patterns of some of the new pieces in Expanded Chess.

4 3 4,1 4,2 4,3 6

No, a traditional 8x8 Chess board is not able to accommodate all the positions for a piece that has 4,Y

or greater L-Type moves.

5 4 5,1 5,2 5,3

5,4 10

6 5 6,1 6,2 6,3

6,4 6,5 15

7 6 7,1 7,2 7,3 7,4 7,5 7,6

21

Combining Components of Movements and Pieces

Now that a thorough analysis to the component movements of the traditional Chess pieces has been presented, we can finally combine these component movements and the all the new pieces shown (Table 3-1) in Expanded Chess into coherent synthesis that is the basis for establishing the value of each new piece.

From the Chapter 2 analysis, there are three fundamental types of movements:

Orthogonal

Diagonal

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L-Type

Each type of movement has magnitude and direction components. Magnitude can be limited or unlimited. The direction of movements can be combinations of forward, backward and sideward. Table 4-3 presents this structure of movements that is used in Expanded Chess. The objective for Expanded Chess is to give the new pieces, standard movements and also to structure the movements for the pieces in an orderly manner using the types and components of movements shown in Table 4-3. It then becomes intuitive what the combined standard component values of the piece are and that the movements of the piece reflect that value.

Table 4-3 Structure of Movements

Type of Movement

Components of Movements

Magnitude Direction

Unlimited Limited Forward Backward Sideward

Orthogonal Yes 3 2 1 Yes Yes Yes

Diagonal Yes 3 2 1 Yes Yes N/A

L-Type N/A 3,2 3,1 2,1 Yes Yes N/A

Let us now assemble the particular combination of types and components of movements for each piece. The resulting Tables 4-4A and Table 4-4B present the movement overview for all the pieces in Expanded Chess. Table 4-4A and Table 4-4B are a tabular representation of Appendix A - Description of Piece & Movement/Coverage Diagram. This tabular representation is necessary to establish the value of each piece on a quantifiable basis. The final step is to determine the factors that influence the value of each piece and then weight these factors and sum up the results to establish the final value of each piece.

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Table 4-4A Combining Movements and Pieces

UNL = Unlimited Movement N/A = Not Applicable

Orthogonal Diagonal L-Type [1] L-Type [2]

Movements Forward Backward Sideward Forward Backward Forward Backward Forward Backward

Piece R C Abbrev.

0 Empress 6 6 EPS UNL UNL UNL UNL UNL 3,2 3,2 N/A N/A

2 Chancellor 6 5 CHL 3 3 3 3 3 3,1 3,1 N/A N/A

1 Emperor 6 4 EPR 2 2 2 1 1 N/A N/A N/A N/A

3 Queen 6 3 QUN UNL UNL UNL UNL UNL N/A N/A N/A N/A

4 Duke 6 2 DUK 3 3 3 3 3 N/A N/A N/A N/A

5 King 6 1 KNG 1 1 1 1 1 N/A N/A N/A N/A

6 Fortress 5 6 FTS UNL UNL UNL N/A N/A 3,2 3,2 N/A N/A

7 Citadel 5 5 CDL 3 3 3 N/A N/A 3,1 3,1 N/A N/A

8 Cathedral 5 4 CTL 2 2 2 N/A N/A 2,1 2,1 N/A N/A

9 Castle 5 3 CSL UNL UNL UNL N/A N/A N/A N/A N/A N/A

b Armory 5 2 ARM 3 3 3 N/A N/A N/A N/A N/A N/A

b Abbey 5 1 ABY 2 2 2 N/A N/A N/A N/A N/A N/A

c Pope 4 6 POP N/A N/A N/A UNL UNL 3,2 3,2 N/A N/A

d Archbishop 4 5 ARB N/A N/A N/A 3 3 3,1 3,1 N/A N/A

e Cardinal 4 4 CRD N/A N/A N/A 2 2 2,1 2,1 N/A N/A

f Bishop 4 3 BSP N/A N/A N/A UNL UNL N/A N/A N/A N/A

g Priest 4 2 PRS N/A N/A N/A 3 3 N/A N/A N/A N/A

h Nun 4 1 NUN N/A N/A N/A 2 2 N/A N/A N/A N/A

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Table 4-4B Combining Movements and Pieces

UNL = Unlimited Movement N/A = Not Applicable *First move only ** Capture only

Orthogonal Diagonal L-Type [1] L-Type [2]

Movements Forward Backward Sideward Forward Backward Forward Backward Forward Backward

Piece R C Abbrev.

i Marshall 3 6 MSL N/A N/A N/A N/A N/A 3,2 3,2 3,1 3,1

j Captain 3 5 CPN N/A N/A N/A N/A N/A 3,2 3,2 2,1 2,1

k Sergeant 3 4 SGT N/A N/A N/A N/A N/A 3,1 3,1 2,1 2,1

l Guard 3 3 GRD N/A N/A N/A N/A N/A 3,2 3,2 N/A N/A

m Archer 3 2 ARC N/A N/A N/A N/A N/A 3,1 3,1 N/A N/A

n Knight 3 1 KNT N/A N/A N/A N/A N/A 2,1 2,1 N/A N/A

o Catapult 2 6 CTP UNL N/A N/A UNL N/A N/A 3,2 N/A N/A

p Ram 2 5 RAM UNL N/A N/A 3 N/A N/A 3,1 N/A N/A

q Mace 2 4 MAC 3 N/A N/A 2 N/A N/A 2,1 N/A N/A

r Lance 2 3 LNC UNL N/A N/A N/A N/A N/A 3,2 N/A N/A

s Hatchet 2 2 HCT 3 N/A N/A N/A N/A N/A 3,1 N/A N/A

t Sword 2 1 SWD 2 N/A N/A N/A N/A N/A 2,1 N/A N/A

u Angel 1 6 AGL UNL 3 UNL UNL N/A N/A N/A N/A N/A

v Saint 1 5 SNT 3 2 3 UNL N/A N/A N/A N/A N/A

w Wizard 1 4 WZD 3 1 3 3 N/A N/A N/A N/A N/A

x Squire 1 3 SQR 3 N/A N/A 3 N/A N/A N/A N/A N/A

y Knave 1 2 KNV 2 N/A N/A 2 N/A N/A N/A N/A N/A

z Pawn 1 1 PWN 2*/1 N/A N/A 1** N/A N/A N/A N/A N/A

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Chapter 5 – Values Of Expanded Chess Pieces

Methods of Establishing Values for Chess Pieces

In order to establish the value of each piece, consider how chess is played. Typically, chess is considered a three-stage game. There is the opening game, the middle game and the end game. The opening part of the game and end game are conceptually simple to define and relatively simple to quantify. This quantification is necessary to determine the value of new chess pieces in Expanded Chess. However, in between the opening and the end game is the middle game. The middle game is more difficult to define conceptually and quantitatively.

The middle game is the stage where the established value of pieces is exploited for trading and capturing pieces. The movements of pieces are limited in the middle game by lack of open squares on the board. However, even with this limited movement, there are still millions of possible combinations for the location of all the pieces.

Now we need to develop an approach to establish the value for Expanded Chess pieces with the realization that there currently no universally accepted method to do so and a new approach needs to be established.

Approach for Establishing Values for Expanded Chess Pieces

The following approach proposed to establish the value of new pieces is an extension of the already empirically existing values of current pieces. The Expanded Chess methodology will break the value of existing pieces into several factors and weight the impact of each factor to establish a summary value for each piece. It is assumed that the value of each piece can be established by determining the range and coverage of each piece at the opening of the game and doing the same at the end of the game when the board is clear. The middle game is then addressed by weighting the range and coverage at the start of the game when the

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location of each piece is always fixed and at the end of the game when the board is clear. The coverage is defined as the number of squares that a piece can attack or defend. The range is defined as the distance (number of squares) that a piece can project its coverage for a given type of movement. Range is linear and the range of an X,Y L-type piece is the magnitude of a X,Y vector. The range and coverage of the movements is divided into types of movements (orthogonal, diagonal and L-Type) and sub-divided into components of movements (magnitude, direction) as shown in Table 4-3. These types of movements and components of movements are used for a more precise weighting. The idea here is that these boundary (initial/final) conditions are well defined and what occurs in between is some weighting of these boundary conditions.

The weighting factors are established empirically by judgment and by results confirmation. This means that the weighting is adjusted until the approach gives a predetermined result in one case and the same weighting will be applied to the other cases. In simple terms, this means that weighting factors are adjusted until the values of the pieces matches the already existing values. In other words, the value of a Queen still needs to be 9 and the value of a Pawn still needs to be 1 for any model of traditional chess.

This approach is obviously not a rigorous analytical process, but that does not mean that it is not acceptable and should not be dismissed as such. A significant advantage of this method is that it can be applied to existing and new pieces, estimates value based on the entire duration of the game and it can be applied to any size of game board, whether it is square or rectangular.

The ultimate objective here is not to establish some new method for refining the existing value of current pieces, which can be used by experienced players to improve their game. The ultimate objective is to develop a method to estimate the value of new chess pieces with results that are directly comparable to the existing values of current pieces without having to wait for the empirical results years of experience to determine the value of new pieces. Players need to know what the value of a new piece is before they start playing a new version of Expanded

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Chess. How would one know how to position and trade pieces without knowing the value of each piece on the board? It would be a frustrating and arbitrary game to play without such knowledge.

Creating a Value System for Expanded Chess Pieces

Start of Game Configuration

At the start of the game, there is an assumed setup for each board size. The following Table 5-1 presents these setups.

Table 5-1 Rows Occupied and Rows Clear at Start of Game

Board Size

Number of

Rows Occupied

by Pieces

Number of

Pieces on the

Board

Number of

Rows Clear

Number of Open

Squares on Board

8x8 4/8

= 50.00% 32

4/8 =50.00%

32

9x9 4/9

=44.44% 36

5/9 =55.56%

45

10x10 4/10

=40.00% 40

6/10 =60.00%

60

11x11 6/11

=54.55 66

5/11 =45.45%

55

12x12 6/12

=50.00% 72

6/12 =50.00%

72

13x13 6/13

=46.15% 78

7/13 =53.85%

91

14x14 8/14

=57.14% 112

6/14 =42.86%

84

15x15 8/15

=53.33% 120

7/15 42.67%

105

16x16 8/16

=50.00% 128

8/16 =50.00%

128

Table 5-1 shows that these setups do not scale upward in an identical manner. There are three ideal board sizes where the rows occupied and the rows clear at the start of the game are equal. These are the 8x8 board, the 12x12 board and the 16x16 board. The other setups end up with either more or less than 50% for the number of rows occupied at the

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start of the game. This variation in the amount of clear space at the start of the game affects the value of the piece for that setup. For example, pieces that are strong in clear space will have a somewhat greater relative value on boards where there is a greater amount of clear space at the start of the game. This effect is taken into account when establishing the value of Expanded Chess pieces on larger board sizes.

Types/Components of Movements Summary Matrix – Appendix D

The first step in creating a value system is to produce a “types/components of movements” summary matrix. The purpose of the matrix is to set up an array which describes all pieces and all the possible types and components of movements available. The array shows which combination of types and components of movements and their magnitude and direction that are used by each piece. The following summary matrix shows the movements available for each piece of the Ruling Class. For example, the Empress (top row across the matrix in Figure 5-1 for a 16x16 game board) shows x‟s for the:

Orthogonal F/B under the 7.50 moves column, (where 7.50 = 15/2 = unlimited moves on 16x16 board)

Orthogonal L/R under the 7.50 moves column, (where 7.50 = 15/2 = unlimited moves on 16x16 board)

Diagonal F L/R under the 7.50 moves column, (where 7.50 = 15/2 = unlimited moves on 16x16 board)

Diagonal B L/R under the 7.00 moves column, (where 7.00 = 14/2 = maximum moves on 16x16 board)

L-1 Type F/LR under the 3,2 moves column,

L-1 Type B/LR under the 3,2 moves column,

L-2 Type F/LR under the 0 moves column,

L-2 Type B/LR under the 0 moves column.

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This matrix represents the combinations of movements for each piece and allows for a systematic approach to distributing the available move combinations for the Empress, Chancellor, Emperor, Queen, Duke and King. The basic premise of move distribution is to create a range of movement capability from highest value to lowest value by systematically reducing the strength and amount of movement. Refer to Appendix D for the complete Types/Components of Movements Summary Matrix.

FiguFigure 5-1 Ruling Class Types/Components of Movements Summary

Matrix

Types/Components of Movements Summary Table – Appendix E

Once the Types/Components of Movements Summary Matrix has been created and the types of movements assigned to each piece, the next step in the process is to determine the value of the range and coverage of each piece based on this matrix. The Types/Components of Movements Summary Table calculates both the coverage and the range for all types of movements for each piece. Once the range and coverage of the piece are determined, the final stage of the process is to evaluate the range and coverage of each piece under different game circumstances. Refer to Appendix D for the complete Types/Components of Movements Summary Table.

Piece R C ABR 7.50 3 2 1 0 7.50 3 2 1 0 7.50 3 2 1 0 7.00 3 2 1 0 4,2 3,2 3,1 2,1 0 4,2 3,2 3,1 2,1 0 4,2 3,2 4,1 3,1 2,1 0 4,2 3,2 4,1 3,1 2,1 0

Empress 6 6 EPS x x x x x x x x

Chancellor 6 5 CHL x x x x x x x x

Emperor 6 4 EPR x x x x x x x x

Queen 6 3 QUN x x x x x x x x

Duke 6 2 DUK x x x x x x x x

King 6 1 KNG x x x x x x x x

Orthogonal F/B Orthogonal L/R Diagonal F/LR Diagonal B/LR L2-Type F/LR L2-Type B/LRL1-Type F/LR L1-Type B/LR

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Weighting/Rating Summary Matrix – Appendix F

The Weighting/Rating Summary Matrix uses the coverage and range of movements from the Types/Components of Movements Summary Table and weights those component coverage and range under different game circumstances as follows:

Coverage at the Start of the Game

Coverage with a Clear Board

Range at the Start of the Game

Range with a Clear Board

The weighting factors are established empirically by judgment and by results confirmation. This means that the weighting is adjusted until the approach gives a predetermined result in one case and the same weighting will be applied to the other cases. In simple terms, this means that weighting factors are adjusted until the values of the pieces matches the already existing values. In other words, the value of a Queen still needs to be 9 and the value of a Pawn still needs to be 1 for any model of traditional chess. The same entire process is repeated for each new game board size to establish the value of the Expanded Chess pieces. Refer to Appendix F for the complete Weighting/Rating Summary Matrix for one game board size.

Value of Expanded Chess Pieces

Table‟s 5-3A and 5-3B present the results of the weighting/rating system evaluation for Expanded Chess. The tables provide the values for each piece in Expanded Chess for 9 different square game board sizes from 8x8 to 16x16. The results are normalized only once with the lowest value piece (the Pawn) taking a value of 1.00. There is no attempt to directly compare the strength of any particular piece at one board size with the same piece at another board size. In simple terms, the value of an

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Empress on an 8x8 board is 12.31 and is 22.97 on a 16x16 board. However, it does not mean that the empress is 22.97/12.31 = approx. 1.9 times as powerful on a 16x16 board as on an 8x8 board. It simply means that am Empress is 12.31 times the value of a Pawn on an 8x8 board and 22.97 times the value of a Pawn on a 16x16 board. Some final comments on Tables 5-3A and 5-3B. Note the values of the traditional players i.e. the Queen, King, Castle, Bishop and Pawn. Since the methodology was mathematical, note that it was not possible to get a perfect balance where all the pieces matched the traditionally assigned values. The values were determined as follows:

The value of the Pawn is set to = 1.00

The value of the Knight is set to = 3.00

The value of the Bishop is set to = 3.00

The value of the Queen is set to = 9.00

The value of the Castle is set to = 6.00 since a Queen – Bishop = Castle (9.00- 3.00 = 6.00)

The value of the King is allowed to float with the Rating Calculation process. The floated value of the King is 3.15 on an 8x8 board.

If one were to quibble at the most theoretical level, it can be argued that the only way for the Bishop and the Castle to be identical in value to the Queen is to have them both occupy exactly the same square at the same time. Since this is not possible in a real game, the value in the chart actually represents an idealized maximum limit, assuming the Bishop and the Castle are occupying the same single square.

When relating the rating of a piece to its row & column index (6 6 to 1 1), pieces with the lower “X” indices generally have lower values the pieces in the upper “X” indices. Values also increase as the “Y” index increases within a sub-classification i.e. either the first three pieces or the second three pieces within in a row. However, the first piece in the second sub-classification might or might not be of greater value than the last piece in the first sub-classification. This is because each sub-classification

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represents a different set of expanded moves. For example, the Castle exceeds the value of the Cathedral for most board sizes.

One final observation is that as the game board gets larger, the spread in value between the Pawn and the Empress increases. This is to be expected since the range and coverage keeps increasing with increasing board size for pieces with unlimited coverage and range such as the Empress. The Pawn is a piece with fixed range and coverage and as the board size increases, its relative influence decreases. Therefore, the net result is a relative increase in the value of the Empress as the size of the board increases and also any other pieces with unlimited moves.

Figures 5-2.1 to 5-2.3, 5-2.4 to 5-2.6 and 5-2.7 to 5-2.9 provide the relative value of each piece in Expanded Chess in order of descending value from maximum to minimum for all 36 pieces at each square board size from 8x8 to 16x16. Although not explicitly calculated, it is expected that the pieces will have similar values on boards that are not square if the total number squares is similar. If the total number of squares is not close to a square board size, it is recommended to interpolate between the values for the piece from the closet two square board sizes.

These bar graphs of Figures 5-2.1 through 5-2.9 show that there are not any significant gaps in the value range of the pieces and only the Empress is of significantly higher value than the next highest value piece. This ratio is 9/5 in Traditional Chess = 1.8 [Value of Queen / Value of Castle]. In Expanded Chess, this ratio varies somewhat as shown in Table 5-2.

Table 5-2 Relative Strength of Queen on Different Board Sizes

Board Size 8x8 9x9 10x10 11x11 12x12 13x13 14x14 15x15 16x16

Empress Value 12.31 14.77 16.32 16.41 17.99 19.69 19.68 21.39 22.97

Next Highest Value Piece

11.81 13.53 13.56 13.14 14.08 15.76 15.92 17.60 19.15

Ratio E/N 1.04 1.09 1.20 1.25 1.28 1.25 1.24 1.22 1.20

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This table shows that although the Empress is powerful, her relative strength is somewhat compressed in Expanded Chess as compared to a Queen in Traditional Chess i.e. her ratio above the second highest value piece ranges from 1.04 to 1.28 and never reaches 1.8 on any game board size as does the Queen‟s ratio over the Castle in traditional chess. One final observation is that the relative values of the pieces can change with the change of the size of the board and the number of rows at the start of the game. Notice how the Queen becomes the second highest value piece once the game board becomes 12x12 or larger. Other pieces also change their relative strength with the change of board size, especially the pieces with unlimited movements.

Now that the values of the Expanded Chess pieces are fully developed, and presented, the next several chapters will establish the approach for expanding other aspects of traditional chess including board configuration and the rules of the game.

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Table 5-3A Value of Expanded Chess Pieces

Game Board Size 8x8 9x9 10x10 11x11 12x12 13x13 14x14 15x15 16x16

Number of Rows Occupied per Side @ Start of Game

2 Rows Occupied 3 Rows Occupied 4 Rows Occupied

Piece R C Abbrev Value

0 Empress 6 6 EPS 12.31 14.77 16.32 16.41 17.99 19.69 19.68 21.39 22.97

2 Chancellor 6 5 CHL 11.81 13.53 13.56 13.14 13.19 13.24 12.93 12.99 13.04

1 Emperor 6 4 EPR 5.19 5.21 5.22 5.11 5.14 5.16 5.07 5.09 5.11

3 Queen 6 3 QUN 9.00 10.68 12.22 12.54 14.08 15.76 15.92 17.60 19.15

4 Duke 6 2 DUK 8.47 9.47 9.48 9.29 9.32 9.34 9.20 9.23 9.25

5 King 6 1 KNG 3.15 3.16 3.16 3.10 3.11 3.11 3.07 3.08 3.08

6 Fortress 5 6 FTS 9.56 11.44 12.53 12.48 13.61 14.72 14.71 15.84 16.96

7 Citadel 5 5 CDL 8.88 10.22 10.27 9.89 9.96 10.02 9.73 9.80 9.86

8 Cathedral 5 4 CTL 7.07 7.09 7.10 6.82 6.85 6.88 6.67 6.71 6.74

9 Castle 5 3 CSL 6.00 7.08 8.16 8.36 9.45 10.53 10.71 11.80 12.89

b Armory 5 2 ARM 5.55 6.15 6.19 6.05 6.09 6.12 6.01 6.05 6.08

b Abbey 5 1 ABY 4.07 4.10 4.13 4.03 4.06 4.08 4.01 4.03 4.05

c Pope 4 6 POP 6.56 7.96 8.43 8.30 8.80 9.42 9.22 9.85 10.34

d Archbishop 4 5 ARB 6.26 7.38 7.37 7.09 7.11 7.12 6.92 6.94 6.96

e Cardinal 4 4 CRD 4.67 4.64 4.62 4.41 4.41 4.41 4.26 4.27 4.28

f Bishop 4 3 BSP 3.00 3.60 4.06 4.18 4.64 5.23 5.22 5.80 6.26

g Priest 4 2 PRS 2.93 3.31 3.29 3.25 3.23 3.22 3.19 3.18 3.18

h Nun 4 1 NUN 1.67 1.66 1.65 1.62 1.62 1.61 1.60 1.59 1.59

EXPANDED CHESS

PAGE 39

Table 5-3B Value of Expanded Chess Pieces

Game Board Size 8x8 9x9 10x10 11x11 12x12 13x13 14x14 15x15 16x16

Number of Rows Occupied per Side @ Start of Game

2 Rows Occupied 3 Rows Occupied 4 Rows Occupied

Piece R C Abbrev Value

i Marshall 3 6 MSL 6.90 8.42 8.44 7.97 8.03 8.09 7.73 7.80 7.86

j Captain 3 5 CPN 6.56 7.34 7.34 6.91 6.96 6.99 6.67 6.72 6.76

k Sergeant 3 4 SGT 6.34 7.05 7.04 6.63 6.67 6.69 6.39 6.43 6.47

l Guard 3 3 GRD 3.56 4.35 4.37 4.13 4.16 4.19 4.01 4.04 4.07

m Archer 3 2 ARC 3.34 4.07 4.07 3.84 3.87 3.90 3.72 3.76 3.78

n Knight 3 1 KNT 3.00 2.99 2.97 2.79 2.79 2.80 2.67 2.68 2.69

o Catapult 2 6 CTP 6.12 7.12 7.73 8.20 8.82 9.44 9.93 10.55 11.17

p Ram 2 5 RAM 5.73 6.42 6.76 6.98 7.33 7.68 7.93 8.29 8.64

q Mace 2 4 MAC 4.66 4.65 4.64 4.53 4.53 4.53 4.46 4.47 4.47

r Lance 2 3 LNC 4.17 4.91 5.26 5.49 5.86 6.22 6.47 6.84 7.20

s Hatchet 2 2 HCT 3.72 4.08 4.09 3.98 4.00 4.01 3.93 3.94 3.96

t Sword 2 1 SWD 2.87 2.86 2.85 2.77 2.77 2.77 2.71 2.72 2.72

u Angel 1 6 AGL 6.69 7.68 8.68 9.35 10.35 11.34 12.01 13.01 14.00

v Saint 1 5 SNT 6.01 6.31 6.61 6.69 6.99 7.28 7.41 7.70 7.98

w Wizard 1 4 WZD 5.73 5.76 5.78 5.61 5.64 5.67 5.54 5.57 5.60

x Squire 1 3 SQR 3.72 3.71 3.70 3.68 3.68 3.67 3.66 3.66 3.65

y Knave 1 2 KNV 2.48 2.47 2.47 2.45 2.45 2.45 2.44 2.44 2.44

z Pawn 1 1 PWN 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

EXPANDED CHESS

PAGE 40

Figure 5-2.1

Value of Pieces in Order of Value

8 X 8 Board 2 Rows Occupied per Player @ Start of Game

12.31 11.81

9.569.00 8.88 8.47

7.07 6.90 6.69 6.56 6.56 6.34 6.26 6.12 6.01 6.00 5.73 5.73 5.55 5.19 4.67 4.66 4.17 4.07 3.72 3.72 3.56 3.34 3.15 3.00 3.00 2.93 2.87 2.481.67

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EXPANDED CHESS

PAGE 41

Figure 5-2.2

Value of Pieces in Order of Value

9 X 9 Board 2 Rows Occupied per Player @ Start of Game

14.7713.53

11.4410.68 10.22

9.478.42 7.96 7.68 7.38 7.34 7.12 7.09 7.08 7.05

6.42 6.31 6.15 5.765.21 4.91 4.65 4.64 4.35 4.10 4.08 4.07 3.71 3.60 3.31 3.16 2.99 2.86 2.47

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EXPANDED CHESS

PAGE 42

Figure 5-2.3

Value of Pieces in Order of Value

10 X 10 Board 2 Rows Occupied per Player @ Start of Game

16.32

13.5612.53 12.22

10.279.48

8.68 8.44 8.43 8.16 7.73 7.37 7.34 7.10 7.04 6.76 6.61 6.19 5.78 5.26 5.224.64 4.62 4.37 4.13 4.09 4.07 4.06 3.70 3.29 3.16 2.97 2.85 2.47

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EXPANDED CHESS

PAGE 43

Figure 5-2.4

Value of Pieces in Order of Value

11 X 11 Board 3 Rows Occupied per Player @ Start of Game

16.41

13.1412.54 12.48

9.899.35 9.29

8.36 8.30 8.20 7.977.09 6.98 6.91 6.82 6.69 6.63

6.05 5.61 5.49 5.114.53 4.41 4.18 4.13 4.03 3.98 3.84 3.68 3.25 3.10 2.79 2.77 2.45

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EXPANDED CHESS

PAGE 44

Figure 5-2.5

Value of Pieces in Order of Value

12 X 12 Board 3 Rows Occupied per Player @ Start of Game

17.99

14.08 13.61 13.19

10.35 9.969.45 9.32 8.82 8.80

8.037.33 7.11 6.99 6.96 6.85 6.67

6.09 5.86 5.64 5.14 4.64 4.53 4.41 4.16 4.06 4.00 3.87 3.68 3.23 3.11 2.79 2.77 2.451.62

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EXPANDED CHESS

PAGE 45

Figure 5-2.6

Value of Pieces in Order of Value

13 X 13 Board 3 Rows Occupied per Player @ Start of Game

19.69

15.7614.72

13.24

11.3410.53 10.02

9.44 9.42 9.348.09 7.68 7.28 7.12 6.99 6.88 6.69 6.22 6.12 5.67 5.23 5.16

4.53 4.41 4.19 4.08 4.01 3.90 3.67 3.22 3.11 2.80 2.77 2.451.61

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EXPANDED CHESS

PAGE 46

Figure 5-2.7

Value of Pieces in Order of Value

14 X 14 Board 4 Rows Occupied per Player @ Start of Game

19.68

15.9214.71

12.9312.01

10.719.93 9.73

9.22 9.207.93 7.73 7.41 6.92 6.67 6.67 6.47 6.39 6.01 5.54 5.22 5.07

4.46 4.26 4.01 4.01 3.93 3.72 3.66 3.19 3.07 2.71 2.67 2.441.60

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EXPANDED CHESS

PAGE 47

Figure 5-2.8

Value of Pieces in Order of Value

15 X 15 Board 4 Rows Occupied per Player @ Start of Game

21.39

17.60

15.84

13.01 12.9911.80

10.559.85 9.80

9.238.29 7.80 7.70

6.94 6.84 6.72 6.71 6.43 6.05 5.80 5.57 5.094.47 4.27 4.04 4.03 3.94 3.76 3.66 3.18 3.08 2.72 2.68 2.44

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EXPANDED CHESS

PAGE 48

Figure 5-2.9

Value of Pieces in Order of Value

16 X 16 Board 4 Rows Occupied per Player @ Start of Game

22.97

19.15

16.96

14.0013.04 12.89

11.1710.34 9.86

9.258.64

7.98 7.867.20 6.96 6.76 6.74 6.47 6.26 6.08 5.60 5.11

4.47 4.28 4.07 4.05 3.96 3.78 3.65 3.18 3.08 2.72 2.69 2.441.59

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EXPANDED CHESS

PAGE 49

Chapter 6 – Size of Game Board

Minimum Board Size

Generally Expanded Chess will be played on game boards larger than 8x8. However, that does not mean traditional chess cannot be expanded in the negative direction (i.e. contracted) to a small game board. For the sake of completeness, a minimum game board size will be developed for Expanded Chess based on contracting a traditional chess game to the lower limit to where it is still implicitly chess. Since the intent of this guide is for any game developed under these guidelines to intuitively have the feel of traditional chess, then the minimum board size should be a 6x6 board. Let‟s explain why this is so. The question implicitly becomes; what is the smallest board size you need to define chess?

Protective Pawns

Each player needs to have a row of Pawns to protect the major pieces at the start of the game. I cannot envision chess without the front row of protective/sacrificial Pawns to give each player time to develop strategy and tactics. Without protective Pawns, the game would immediately start off with major pieces overexposed and attacked in a war of attrition with no ability to develop any strategic positional advantage.

Another problem is that White could place Black in check on the opening move if there were not protective Pawns. This seems like a huge advantage for White and could not be permitted if trying to maintain the fairness of the game. Since Pawns are necessary, the next item becomes how much space is needed between the Pawns of each side. Let‟s review the three most probable options.

EXPANDED CHESS

PAGE 50

Four Rows Deep

If you select a board that in 4 rows deep in total as shown in Figure 6-1, then each player occupies two rows and there is no space between the Pawns. There is a serious problem with this approach. The board would be locked up with Pawns facing each other head to head at the start of the game. The only way to start the game would be to capture the opponent‟s piece on your opening move. This does not seem right. In Chess, every piece is free to make at least one move before being captured and this implicit assumption should be maintained in any version of chess so the board cannot be 4 rows deep.

f 3 533

9 z z z z z z z z f 3 5

3 9

Figure 6-1 Four Rows Deep Board – No Room for Pawns to Move

Five Rows Deep

If the board is 5 rows deep, as is shown in Figure 6-2, then there is one open row between pieces at the start of the game. In this case, a Pawn would be entitled to a first move. However, that move would immediately put the Pawn in a position to be captured. Again this is does not seem to match the intent of traditional chess. Each player should be able to make an opening move that does not automatically leave that piece open to capture. Therefore, a five row deep board as the minimum is not acceptable.

EXPANDED CHESS

PAGE 51

9 3 5 f

3

9 z z z z z z

z z z z 9 3 5 f

3 9

Figure 6-2 Five Rows Deep Board – Pawns are Threatened on Opening Move

Six Rows Deep

As a result of the aforementioned discussion, the next size for a board is 6 rows deep as is shown in Figure 6-3. This size allows each player to make an opening move and not be immediately subject to capture. Granted, the Pawn is allowed to move only one row forward but that can be the case in traditional chess as well.

9 f 3 5 n 9 z z z z z z z

z z z z z 9 f 3

33

53

n 9

Figure 6-3 Six Rows Deep Board – Pawns can be safe on opening move if they move only one square ahead

EXPANDED CHESS

PAGE 52

Minimum Board Width

Once the minimum depth of the board has been set at 6 rows, the next question becomes: how wide should the board be? This question really becomes: what is the minimum number of pieces on the board that are acceptable? In order to retain the intuitive feel of tradition chess, the simple answer is to have at least one of each type of piece on the board at the start of the game. This requires a minimum width of 5 columns. You need Pawns in front row and a King, Queen, Knight, Bishop and Castle on the back row. However, to preserve some semblance of symmetry, there should be two castles. Therefore, the back row arrangement is as shown in Figure 6-4:

z z z z z z 9 f 3 5 n 9

or

z z z z z z 9 n 3 5 f 9

Figure 6-4 Minimum width of the game board – two possible arrangements.

Either way is acceptable for Expanded Chess and could be an option for each player at the start of the game.

To summarize, the minimum size of the chess board should be 6x6 with at least one of each type of piece at the start of the game.

EXPANDED CHESS

PAGE 53

Maximum Size of Board

In principle, there is no maximum board size. But, in reality, the maximum board size should be constrained for practical reasons. A large board requires a large number of pieces at the start of the game. An 8x8 chess board with each player‟s back two rows filled with pieces requires 2 x 8 x 2 = 32 pieces. If the board is 16x16 then to keep the board half full of pieces at the start of the game (same ratio as the 8x8 board) then 4 x 16 x 2 = 128 pieces which is 4 times as many as the 8x8 board. A very large number of pieces have the following drawbacks:

It will be difficult to recall the position of each piece at the start of the game to set up the board.

It will be difficult to recall the moves of each piece

It will take many moves to establish a strong attack and the game will progress at glacial pace.

It will be difficult to envision moves very many levels in to future because there are so many possible combinations to keep track of in one‟s mind.

Eventually a large board with many pieces will become too cumbersome and intimidating to enjoyably play chess. So where should the upper limit be? It seems like an arbitrary decision. If you combine four 8x8 boards you will get one 16x16 board. This is a reasonable upper limit for Expanded Chess. Now that we have established the boundaries of the size of the Expanded Chess board, next we will examine the arrangement of the game board in the next chapter.

EXPANDED CHESS

PAGE 54

Chapter 7 – Arrangement of Game Board

The traditional chess game board is an array of 8 squares wide by 8 squares deep. The fundamental issue for board arrangement is how many new dimensions to add to the game. There are four significant possibilities including:

the shape of the board

the shape that each piece rests on

the number of dimensions of the board

the number of colors used on the board.

Let us explore the options available for each one of the aforementioned possibilities.

Shape of the Game Board

Square and Rectangle Shapes

Does the board need to remain perfectly square? In lieu of being perfectly square, the board can be a rectangle with a reasonable aspect ratio. Table 7-1 presents the recommended aspect ratio (highlighted) range for the game board.

If the game board was to remain perfectly square (XxY, X=Y) then there are only 11 board sizes available for Expanded Chess as shown in Table 7-1 as the cells on the main diagonal. If the aforementioned rectangular ratios are accepted then there are 15 more game board sizes available for use as is shown in Table 7-1 as the cells highlighted which are off the main diagonal. It can be envisioned that some experimentation will reveal that, as the boards are expanded and the rows occupied at the start of the game increase, a rectangular board might be preferred.

EXPANDED CHESS

PAGE 55

Table 7-1 Recommended Aspect Ratios for the Game Board

Board Width (No. of Squares)

6 7 8 9 10 11 12 13 14 15 16

Board Depth (No. of

Squares)

Recommended Aspect Ratio of Game Board [W/D]

6 1.00 1.17 1.33 1.50 1.67 1.83 2.00 2.17 2.33 2.50 2.67

7 0.86 1.00 1.14 1.29 1.43 1.57 1.71 1.86 2.00 2.14 2.29

8 0.75 0.88 1.00 1.13 1.25 1.38 1.50 1.63 1.75 1.88 2.00

9 0.67 0.78 0.89 1.00 1.11 1.22 1.33 1.44 1.56 1.67 1.78

10 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60

11 0.55 0.64 0.73 0.82 0.91 1.00 1.09 1.18 1.27 1.36 1.45

12 0.50 0.58 0.67 0.75 0.83 0.92 1.00 1.08 1.17 1.25 1.33

13 0.46 0.54 0.62 0.69 0.77 0.85 0.92 1.00 1.08 1.15 1.23

14 0.43 0.50 0.57 0.64 0.71 0.79 0.86 0.93 1.00 1.07 1.14

15 0.40 0.47 0.53 0.60 0.67 0.73 0.80 0.87 0.93 1.00 1.07

16 0.38 0.44 0.50 0.56 0.63 0.69 0.75 0.81 0.88 0.94 1.00

Aspect Ratio = Board Width (No. of Squares) / Board Depth (No. of Squares)

As an example, consider a board that is 14 rows wide with 4 rows occupied per side at the start of the game. There are already 3 rows clear in the middle of the board per side (or 6 rows total) at the start of the game. Depending on the selection of the front row pieces, it might be advantageous to remove two rows from the game board so that there is less space to maneuver during the game, which will speed up play on a large board. If this were the case, then the game board would be 14 columns wide x 12 rows deep.

Board Arrangement and Notation

Expanded Chess uses the same approach as traditional chess for board arrangement and notation. Let us begin with an 8x8 board with standard FIDE algebraic notation as shown in Figure 10-1. The notation is based on the bottom left corner of the board being a dark square and being the origin of the board. This square is labeled “a01”. The rows are numbered upward originating from the bottom row and labeled as “01” to 08”. The columns are lettered rightward originating from the left most column and labeled “a” to “h”.

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08 9 n f 3 5 f n 9 07 z z z z z z z z 06

05

04

03

02 z z z z z z z z 01 9 n f 3 5 f n 9

a b c d e f g h

Figure 7-1 8x8 Game Board with

Standard FIDE Algebraic Notion

This same convention is used in Expanded Chess. The bottom left square of the board will always be a dark square and will always be the origin, regardless of the size of the board and regardless of whether the board is even, odd or mixed in size. As an example, for a 15x15 board as is shown in Figure 7-2:

the rows will be numbered from the bottom row starting with “01” and continuing upward to “15”

the columns will be lettered from the right column starting with “a” continuing leftward to “o”.

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15 o15

14

13

12

11

10

09

08 g08

07

06

05

04

03

02

01 a01

a b c d e f g h i j k l m n o

Figure 7-2 15x15 Game Board with

Expanded Chess Algebraic Notion

Locations a01, g08 and o15 are shown

Other Shapes

The shape of the board and the shape that each piece rests on (normally a square) depend on whether or not there are more than two sides playing the game. What is the relationship between these two? If there are only two sides playing simultaneously, then there is no need for more elaborate geometric shapes for the board or the resting shapes than the traditional board. The two sides can face each other directly at the start of the game with use of the traditional square board. However, if there were three sides playing, the geometry of the game board would need to reflect this by having some type of triangular or hexagonal shaped board which can literally accommodate three sides facing each

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other in a symmetrical manner at the start of the game. This would also result in the shape of the board “squares” being modified to be either triangles or hexagons to create the three-way symmetry required. In order for each side‟s opening positions not to interfere directly on one of the other sides‟, it would require a hexagonal-shaped board where each side occupied alternating segments of the hexagon. It can be similarly envisioned that four-sided Expanded Chess would require an octagonal shaped board to accommodate the four sides without each side interfering with the two closest other sides opening position if one were to attempt to use a square board. Looking one step past the geometric considerations, if Expanded Chess were to attempt to entertain having more than two sides playing simultaneously, the fundamental nature of the game is altered because a two-sided game is always a head-to-head confrontation with direct elimination. Once the game has more than two sides, other elimination scenarios become possible and necessary to determine a winner. With three sides or more, the stronger sides can become allies and eliminate weaker sides and then go head-to-head in the last phase of the game. It is also possible to rate sides as to the order of their finish by measuring each side‟s performance, relatively, against the other sides or against some predetermined and fixed standard. The resulting change of the fundamental nature of a three-or-four-sided game (away from a head-to-head confrontation with direct elimination) leads to the conclusion that these changes are inherently too radical for Expanded Chess and are left beyond the scope of Expanded Chess. Therefore, the shape of the game board should remain four sided and other shapes are not considered. However in a two sided game, each side can have more than one player. Refer to Chapter 8 for a discussion concerning the number of players per side in Expanded Chess.

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Number of Spatial Dimensions of the Board

There are two possibilities for the number of spatial dimensions. The game can remain as a two dimensional board game as it has been for hundreds of years or it can add the third spatial dimension and become like a prop from Star Trek.

On the spatial dimensionality issue, Expanded Chess is remaining uncharacteristically conservative and traditional. Chess, even in an Expanded Chess format, should remain a two dimensional game. The largest problem with adding a third dimension is that it is not entirely intuitive how to expand the game into another dimension. There are several road blocks conceptually. Do you play several games simultaneously that are semi-independent, or do you allow for pieces on one board level to move up or down to another level? If the different levels are semi-independent what relationship ties these levels together? If you allow movement between levels, how do you capture pieces on another level? How do you position your King/Emperor and other pieces in a manner that is analogous to a 2-D board so that the move to 3-D chess can be understood in a relatively straightforward manner by an inexperienced 2-D chess player? There are so many possibilities here. Most rule extensions and the nature of play will not be intuitively analogous based on 2-D chess. This will lead to confusion and frustration and will negate the main premise of Expanded Chess that expansion should be intuitively based on current rules and nature of play.

There is also the problem of the physical awkwardness of setting up and playing a 3-D Chess game in the physical world. You need to place your hand into a 3 dimensional cell to move your pieces without knocking over other pieces or the game cube for each move. The setup, space requirements, maintenance and take down of such a game cube is also a significant nuisance factor working against the quick general acceptance of such a game. In the future holographic technology will

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address this issue and 3-D chess in the physical world will become more viable.

The other way to approach 3-D chess is to play such a game on a computer. The game would consume a significant amount of computing power to run the 3-D game engine and require a large monitor with high resolution graphics to clearly display all the pieces at the different locations in 3-D space. The game would need to be viewed from continuously varying perspectives as each player attempted to develop their next move and might be beyond the capabilities of a home computer for the near future. All issues considered, 3-D chess is the subject of a completely separate book and is currently beyond the scope of Expanded Chess.

Number of Colors Used on the Board

Traditional chess has two colors in an alternating pattern of light colored squares and dark colored squares. Each color must not touch another square of its own color orthogonally (either in forward/backward or sideward direction). Using a two dimensional board, it is surprisingly difficult to achieve the simple symmetry of a two dimensional, square, two color board with any other number of colors than two. Let us examine the requirements of using more than a two-color game board by attempting to develop a three and four-color game board.

Three Color Board

The three color board is problematic right out of the gate. When you take three colors and attempt to use them to create a meaningful pattern, the first problem you notice is that by alternating three colors you automatically have an aesthetic necessity for board sizes that are multiples of 3, i.e. 6x6, 6x9, 9x9, 9x12, 12x12, 12x15, and 15x15. There is a

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further limitation of the configuration of the game board. The square board sizes are limited to 6x 6, 9x 9, 12x12, and 15x15.

The next noticeable issue that you encounter is having many possible ways to alternate the colors and create a pattern. A first inclination is to try the single right sliding pattern. Let‟s assume that we use RED (R), GREEN (G) and BLUE (B) as our three colors. The first row of a 9x9 board will be as shown in Figure 7-2a:

R G B R G B R G B

Figure 7-2a –One Row To create the second row, the red square is slid one position to the right and lines up under the green square in the top row as shown in Figure 7-2b:

R G B R G B R G B

B R G B R G B R G

Figure 7-2b - Two Rows To create the third row, the red square is slid one position to the right and lines up under the green square in the second row as shown in Figure 7-2c:

R G B R G B R G B

B R G B R G B R G

G B R G B R G B R

Figure 7-2c – Three Rows

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To complete the 9x9 board, the 3x9 pattern above is repeated twice more below the first pattern as Figure 7-2d:

R G B R G B R G B

B R G B R G B R G

G B R G B R G B R

R G B R G B R G B

B R G B R G B R G

G B R G B R G B R

R G B R G B R G B

B R G B R G B R G

G B R G B R G B R

Figure 7-2d – 9x9 Board

The obvious problem with this pattern is that the diagonals are a single color in one direction and three alternating colors in the opposite direction. Attempting to correct this major deficiency is not possible.

The most symmetric arrangement developed using three colors was the following board shown in Figure 7-2e:

B G R B R B R G B

G R G R G R G R G

R B R B R B R B R

B G B G B G B G B

G R G B G R G B G

R G R G R G R G R

B R B R B R B R B

G B G B G B G B G

R B B R B R B G R

Figure 7-2e Improved 9x9 Board

There are multiple patterns that define the diagonals. Some patterns are symmetrical; some patterns are anti-symmetrical. The longest run of squares of the color on a diagonal is three. In the end, there is no possible way to configure a board with diagonals with the same color from one end of the board to the other end in both directions using three colors. This exercise seems to be as complicated as color theory in

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particle physics. Common sense shows that there is no way that you can make this color arrangement work as a scheme for traditional or Expanded Chess.

Four Color Board

The four color board is also challenging right out of the gate. When you take four colors and attempt to use them you automatically have an aesthetic preference for board sizes that are multiples of 4, i.e. 8x8, 8x12, 12x12, 12x16, and 16x16. There is a further limitation of the configuration of the game board. The square board sizes are limited to 8x 8, 12x12 and 16x16.

Again there are many possible ways to alternate the colors and create a pattern. Again we try the single right sliding pattern. Let‟s assume that we use BLUE (B), GREEN (G) and RED (R) and YELLOW (Y) as our four colors. The first row of an 8x8 board is shown in Figure 7-3a:

B G R Y B G R Y

Figure 7-3a - One Row

To create the second row, the blue square is slid one position to the right and lines up under the green square in the top row as shown in Figure 7-3b :

B G R Y B G R Y

Y B G R Y B G R

Figure 7-3b - Two Rows

To create the third row, the blue square is slid one position to the right and lines up under the green square in the second row as shown in Figure 7-3c:

B G R Y B G R Y

Y B G R Y B G R

R Y B G R Y B G

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Figure 7-3c - Three Rows

To create the fourth row, the blue square is slid one position to the right and lines up under the green square in the third row as shown in Figure 7-3d:

B G R Y B G R Y

Y B G R Y B G R

R Y B G R Y B G

G R Y B G R Y B

Figure 7-3d - Four Rows

To complete the 8x8 board, the 4x8 pattern above is repeated once more below the first pattern as shown in Figure 7.3e:

B G R Y B G R Y

Y B G R Y B G R

R Y B G R Y B G

G R Y B G R Y B

B G R Y B G R Y

Y B G R Y B G R

R Y B G R Y B G

G R Y B G R Y B

Figure 7-3e – 8x8 Board

Again the problem with this pattern is that the diagonals are a single color in one direction and various alternating colors in the opposite direction.

After some deliberation and experimentation, the most symmetric arrangement developed using four colors was the following board shown in Figure 7-3f:

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B G B G B G B G

Y R Y R Y R Y R

B G B G B G B G

Y R Y R Y R Y R

B G B G B G B G

Y R Y R Y R Y R

B G B G B G B G

Y R Y R Y R Y R

Figure 7-3f – Improved 8x8 Board

This pattern can be described as a four-color, dual-weave pattern. Each color appears to be woven into the pattern in a symmetrical manner. Beginning in the top-left corner of the board, the colors alternate in one pattern downward; blue/green the first row then and then yellow/red second row and then the pattern repeats. The colors alternate in one pattern sideward; blue/yellow the first column then and then green/red second row and then the pattern repeats. For the game board blue/red alternate along the diagonal from top left to bottom right and green/yellow alternate along the diagonal from top right to bottom left.

This four-color, dual-weave pattern is be considered an acceptable game board configuration if one were to relax the restriction along the diagonal pattern being a single color in one direction and the other color along the opposite diagonal and also allow two alternating colors in one diagonal direction and the other two alternating colors in the other diagonal direction. If one were to redefine the diagonal configuration as one half of the colors alternating along one diagonal direction and the other half of the colors alternating along the opposite diagonal direction then both a two color game board and a four color game board are acceptable. The two-color, traditional chess game board becomes a slightly special case where having only one color along each diagonal direction results in not being able to alternate colors along a given diagonal.

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This can be expressed as: a = alternates colors c = no. of colors for c = 2 , a = 0 and for c = 4, a = 2

A four color game board can be considered to control the initial placement of pieces on a large game board or limit the movements of certain pieces. This type of board can also be used for games with two players per side as a mechanism to inherently control which squares each player can make moves from during the game. The next chapter considers playing Expanded Chess with more than one player per side.

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Chapter 8 – Number of Players During Game

Traditional Chess

In traditional chess there are two players who are opponents at the start of the game. The game starts with, plays all the way through and ends with these same two opponents. There are no provisions in the rules of traditional chess for making any changes to this original arrangement.

Expanded Chess

In Expanded Chess there will two players who are opponents at the start of the game generally. However, because of the size of the game board extends up to 16x16, which will have four times the number of squares as an 8x 8 game board, it is easily conceivable to have two, three or even four players playing for one team in Expanded Chess. Table 8-1 presents the minimum size of game board recommended based on the assumption that in traditional chess the game board is 8x8 and each player has a 32 square domain to watch over (one-half the game board).

If one were to maintain the 32 square domain as a guideline as the board size increases, Table 8-1 presents the minimum size of game board needed to support in increase in the number of players. As an example, three players per side would need 3 x 32 = 96 squares per side. A game board sized 14x14 = 196 squares. Each side has a domain of 196/2 = 98 squares. Finally 98/3 = 32.67 squares as the starting domain per player. Therefore, the 14x14 or larger game board is recommended size for 3 players per side.

In traditional chess each side has 16 pieces at the start of the game. If the number of players per side is increased at the start of the game, in Expanded Chess, then it is possible that the number of pieces can be increased at the start of the game in proportion to the number of players. Table 8-1 also provides the recommended number of pieces per side as

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the number of players and game board size is increased. Continuing with the previous example, a side with three players would use a recommended 14x14 game board and this would result in the use of 3 row x 14 pieces per row = 42 pieces.

Table 8-1 Minimum Size of Game Board and Number of Pieces Recommended for Multiple

Players Per Side

Number of Players Per Side

Ideal Number

of Squares

Actual Number

of Squares

Minimum Recommended Size of Game

Board

Starting Domain

Per Player

(squares)

Actual Pieces

per Domain

Ideal Pieces

per Domain

1 Player per Side

64 64 8x8 32.00 2 x 8 = 16 16

2 Players per Side

128 121 11x11 30.25 3 x 11 = 33 32

3 Players per Side

192 196 14x14 32.67 3 x 14 = 42 48

4 Players per Side

256 256 16x16 32.00 4 x 16 = 64 64

Once multiple-player teams are introduced into Expanded Chess, multiple moves per turn are also recommended so that each player can participate during each turn for a side. It would not be exciting if there was still only one move per turn with multiple players and each player had to wait one, two or three turns to make a move. Increasing the number of moves per turn introduces an exciting new dynamic to the game and expands the combinoric potential to challenge any Expanded Chess computer program. As a result of this addition, several new parameters can be added to the game. These new parameters include:

limitations of each player‟s access to some area on the game board

type of moves per turn including: o multiple moves with different piece each move o multiple moves with same piece each move o multiple pieces with any piece each move

limitations of each player‟s access to pieces

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Table 8-2 presents the possible combinations of movements per turn when there are one to four players per side for an Expanded Chess game. The table provides the recommended combinations of players per side for the number of moves per players with aforementioned parameters. There is potential, especially on a large game board, to assign each player a zone of participation, i.e. the designated player must make all moves into the designated area. For example, it can be envisioned that one player on the team will be a defending specialist cover the back half of the board while other player on the team will an attacking specialist covering the front half of the board. It is also possible that each player could make all the moves on one color square. This game would take place on a four color board with four players. There is one weakness with this type of approach. A certain zone or color of square might be intentionally avoided by one team to limit a strong player„s ability to participate in the game and leaving all the move to the weaker players on the other team. There is also potential to increase the number of types of moves per turn. This includes multiple moves with a different piece each move. If there were three players on one side and each one were to have a move, then each one would be required to move a different piece within the turn such as an Empress, a Bishop and a Duke. It is also possible to include multiple moves with the same piece each turn. With this approach, it would be possible to move a Queen or any other piece twice or more in succession in a single turn without the opposition moving and would make the game much more dynamic and produce large momentum shifts. Finally, the two approaches already discussed can be combined together so that multiple moves with any piece for each move can be used during one turn. For a four move turn, a sequence of moves such as Cardinal-Knight-Knight-Cardinal would be acceptable.

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There is also potential to allow each player to have access to move only certain pieces during the game. This controlled access could be based on the type of the piece that each player is able to move. This assignment of pieces can be based on the value of the piece such that the senior player moves the higher value pieces while the junior player moves the lower value pieces. Other possibilities include each player having an equivalent group of pieces to move. As an example, each player might be assigned one half the pieces, each one controlling an equal number of each type of piece. The King or Emperor would either be assigned to both players as a team or to the senior player on the team.

Table 8-2 Possible Combinations of Moves on Per Turn Basis with Multiple Players per Side

Number of moves per turn 1 2 3 4

Number of Players per Side

Number of Players per Side

Number of Players per Side

Number of Players per Side

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Parameter A B C

Limits

A Player’s

access to some area

of game board

None X N/A N/A N/A X X N/A N/A X N/A X N/A X X N/A X

Limited N/A N/A N/A N/A N/A X N/A N/A N/A N/A X N/A N/A X N/A X

B Type of

Moves per turn

Multiple Moves with

Different Piece Each

Move

N/A N/A N/A N/A X X N/A N/A X N/A X N/A X X N/A X

Multiple Moves with

Same Piece Each Move

N/A N/A N/A N/A X X N/A N/A X N/A X N/A X X N/A X

Multiple Moves with Any Piece

Each Move

N/A N/A N/A N/A X X N/A N/A X N/A X N/A X X N/A X

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Table 8-2 Possible Combinations of Moves on Per Turn Basis with Multiple Players per Side

Number of moves per turn 1 2 3 4

Number of Players per Side

Number of Players per Side

Number of Players per Side

Number of Players per Side

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Parameter A B C

Limits

C Player’s

access to pieces

None X N/A N/A N/A X X N/A N/A X N/A X N/A X X N/A X

Limited N/A N/A N/A N/A N/A X N/A N/A N/A N/A X N/A N/A X N/A X

X = Available combinations

N/A = Not available combinations

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Chapter 9 – Types and Number of Pieces at Start of Game

This chapter discusses the approach developed to determine the number and type of playing pieces at the start of the game as the size of the game board changes.

When a chess game is started, there are a fixed number, type and arrangement of playing pieces on the game board. As the game board size increases, so should the number of playing pieces also increase. There are several different items which should be considered to place a reasonable number and type of pieces on the game board at the start of a game.

Open Rows

The first item for consideration is that the number of open rows at the start of the game must be sufficient so each player can make at least one opening move with each piece on the front row (nearest to the opposing pieces), before that piece is threatened with capture. This situation is addressed by the proper selection of the movement/capture range of the pieces on the front row and the number of open rows at the start of the game.

Defining Low, Intermediate and High Level pieces

The next item for consideration is the distribution of the types of pieces as determined by a type level classification for the playing pieces. This type classification is necessary because it becomes obvious after playing some Expanded Chess games that placing High level pieces in precarious locations at the start of the game is not done in traditional chess and makes no sense in Expanded Chess either. Low level pieces should not be able to overwhelm and dispense with impunity

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Intermediate and High level pieces near the start of the game because of lack of thought about the placement of pieces at the start of the game by the Expanded Chess game developer. The Intermediate, High and Checkmate level pieces should be placed on the game board in such positions that they are protected from attack by Low level opponents until they move aggressively toward the open central portion of the game board.

The primary basis for level classification is the range of movement of the type of piece. Low level pieces are restricted to one, two or three moves straight or diagonally forward. Intermediate and High level pieces do not have this restriction. Intermediate pieces are classified as having fixed range in all directions or unlimited range in the forward direction and limited range in other directions. High level pieces have an unlimited range in more than one direction. Low level pieces are needed in the first row to ensure that each piece can make one move before being attacked by an opponent‟s piece. The Intermediate and High level pieces are placed behind the Low level pieces and are to be protected at the start of the game.

Table 9-1 presents the classification level of all the pieces in Expanded Chess and divides them into Low, Intermediate, High and Checkmate levels. The table also defines each level and places each type of piece in its level for determining its location at the start of the game.

Table 9-1 Level of Pieces @ Start of Game

Level Definition of Level Pieces in Level

Low Forward moving only limited range, one, two or three moves straight ahead or diagonal, Promotion to another piece after achieving a defined amount of forward motion.

Squire (1 3)

Knave (1 2)

Pawn (1 1)

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Table 9-1 Level of Pieces @ Start of Game

Level Definition of Level Pieces in Level

Intermediate L Pieces with a limited range in all directions.

Duke (6 2)

Chancellor (6 5)

Citadel (5 5)

Cathedral (5 4)

Armory (5 2)

Abbey (5 1)

Archbishop (4 5)

Cardinal (4 4)

Priest (4 2)

Nun (4 1)

Marshall (3 6)

Captain (3 5)

Sergeant (3 4)

Guard (3 3)

Archer (3 2)

Knight (3 1)

Ram (2 5)

Mace (2 4)

Hatchet (2 2)

Sword (2 1)

Wizard (1 4)

L&U1 Pieces with an unlimited range in the forward direction and limited range in other directions.

Lance (2 3)

Saint (1 5)

High Pieces with an unlimited range in more than one direction.

Empress (6 6)

Queen (6 3)

Fortress (5 6)

Castle (5 3)

Pope (4 6)

Bishop (4 3)

Catapult (2 6)

Angel (1 6)

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Table 9-1 Level of Pieces @ Start of Game

Level Definition of Level Pieces in Level

Checkmate Pieces that when captured end the game with victory for the player who captures the piece.

Emperor (6 4)

King (6 1)

Table 9-1 shows that there are 3 types of pieces in the Low level, 23 types in the Intermediate level, 8 types in the High level and 2 types in the Checkmate level for a total of 36 types of pieces in Expanded Chess. Also from Table 9-1, traditional chess has 1 Low level type (Pawn), one Intermediate level type (Knight), 3 High level types (Queen, Castle and Bishop) and one Checkmate level type (King). The challenge here for Expanded Chess is to increase the number of types at each level at the start of the game and increase them in an orderly manner that preserves the feel of traditional chess while making best use of the increased number of types of pieces in each level and enriching Expanded Chess.

In order to increase the number of types of pieces in an orderly manner, as the game board gets larger, several factors are considered. These factors which affect the game board at the start of the game include:

total number of levels

total number of the types of pieces

total number of pieces

number of types of pieces in each level

number of pieces in each level

The aforementioned factors have several limitations. Minimums and maximums should be considered for these factors. For a small, medium or large game board, all four levels of types of pieces should be included in Expanded Chess. Let us review each of these factors for their limitations.

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Scaling Up

There are two parameters to be scaled up as the size of the game board increases. These two parameters include the number of types of pieces and the number of pieces.

Scaling Up Types of Pieces

The total number of types of pieces can vary significantly at the start of the game, depending on the size of the board. At the small end, on a traditional chess board (8x8), there are 6 types of pieces. For Expanded Chess, on a large board (16x16), the number of types of pieces is likely most constrained by human memory. It would be hard to imagine a game with more than 24 types of pieces, which is 4 times as many as traditional chess. Recalling all the moves and all the potential interactions of more than 24 types of pieces would be extremely, mentally taxing for even for the world‟s most brilliant chess players. Between these two limits of the types of pieces, there needs to be a scaling up of the numbers to be used for each size of game board. Table 9-2 provides such a matrix for scaling up the number of levels as the game board size increases.

Table 9-2 Recommended Minimum Number of Types of Pieces

per Levels per Side @ Start of Game

Board Size Low

Inter-mediate

High Check- mate

Level Total

Small Board 2 Rows occupied per side at start of game

8x8 1 1 3 1 6

9x9 1 2 3 1 7

10x10 1 3 3 1 8

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Table 9-2 Recommended Minimum Number of Types of Pieces

per Levels per Side @ Start of Game

Board Size Low

Inter-mediate

High Check- mate

Level Total

Medium Board 3 Rows occupied per side at start of game

11x11 2 3 4 1 10

12x12 2 5 4 1 12

13x13 2 7 4 1 14

Large Board 4 Rows occupied per side at start of game

14x14 3 9 5 1 18

15x15 3 10 6 1 21

16x16 3 12 7 2 24

The number of types of pieces in the Low level should be one on a small game board (8x8 to 10x10), two on a medium sized game board (11x11 to 13x13) and three on a large sized game board (14x14 to 16x16). The number of types of pieces in the Intermediate level should be one to three on a small game board (8x8 to 10x10), three to seven on a medium sized game board (11x11 to 13x13) and nine to twelve on a large sized game board (14x14 to 16 x16). The number of types of pieces in the High level should be three on a small game board (8x8 to 10x10), four on a medium sized game board (11x11 to 13x13) and five to seven on a large sized game board (14x14 to 16 x16). The number of types of pieces in the Checkmate level should be one on a small game board (8x8 to 10x10), one on a medium sized game board (11x11 to 13x13) and one or possibly two on a larger sized game board (14x14 to 16 x16) if there is some scheme to require a capture of two pieces for checkmate.

Scaling Up Number of Pieces

The total number of pieces can vary significantly at the start of the game, depending on the size of the board. At the small end, on a traditional chess board (8x8), there are 16 pieces per side. The maximum total number of pieces in Expanded Chess, based on a maximum 16x16 board size, is 64 pieces per side, with the complete utilization of back four rows

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for each side of the board. Between these two limits of the number of pieces, there should be a scaling up of the number of pieces used for each size of game board.

Tables 9-3A, B & C present the recommended number of pieces per side for each level, low, Intermediate, High and Checkmate level. Table 9-3 is developed with the awareness that there are more types of Intermediate pieces (23 of 36 total types) than any other type of piece. After the number of other pieces from other levels is set, the Intermediate number is based on filling up the remaining spaces on the game board. The number of Low level pieces is generally increased with the number of rows because the first row always must be occupied by Low level pieces. On larger game boards consideration can be given to adding extra Low level players in the second row to add variety and enrich Expanded Chess. The number of High level players is generally increased with the number of rows and is maintained as about one third to almost one half of all the pieces on the board.

Table 9-3A Recommended Number of Pieces

per Levels per Side @ Start of Game

Board Size Low

Inter-mediate

High Check mate

Level Total

Small Board 2 Rows occupied per side at start of game

8x8 Minimum 8 2 5 1 ---

8x8 Normal 8 2 5 1 16

8x8 Maximum 8 2 5 1 ---

9x9 Minimum 9 2 4 1 ---

9x9 Normal 9 2 6 1 18

9x9 Maximum 9 4 6 1 ---

10x10 Minimum 10 2 5 1 ---

10x10 Normal 10 2 7 1 20

10x10 Maximum 10 4 7 1 ---

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Small Board - Table 9-3A

On a small game board, with two rows occupied, the front row needs to contain Low level pieces. The minimum number of pieces is 8 and is based on the front row of an 8x8 board being occupied with Low level pieces. The maximum number of Low level pieces is 10 and is based on a 10x10 board with the front row being occupied with Low level pieces. The minimum number of Intermediate level pieces is 2 based on an 8x8 board being occupied with traditional chess pieces (Knights). The maximum number of Intermediate level pieces is 4 and is based on a 10x10 board. The minimum number of High level pieces is 5, based on an 8x8 board being occupied with traditional chess pieces (Bishops, Castles and Queen). The maximum number of High level pieces is 7 and is based on a 10x10 board with Intermediate level pieces at their minimum.

Table 9-3B Recommended Number of Pieces

per Levels per Side @ Start of Game

Board Size Low

Inter-mediate

High Check mate

Level Total

Medium Board 3 Rows occupied per side at start of game

11x11 Minimum 11 6 7 1 ---

11x11 Normal 11 6 15 1 33

11x11 Maximum 11 14 15 1 ---

12x12 Minimum 12 6 11 1 ---

12x12 Normal 12 12 11 1 36

12x12 Maximum 12 12 17 1 ---

13x13 Minimum 13 12 9 1 ---

13x13 Normal 13 16 9 1 39

13x13 Maximum 13 16 19 1 ---

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Medium Board – Table 9-3B

On a medium sized board with three rows occupied, at least the first row will need to be Low level pieces as well. The minimum number of pieces is 11 based on the front row of an 11x11 board being occupied with Low level pieces. The maximum number of Low level pieces is based on a 13x13 board with the front row being occupied with Low level pieces. The minimum number of Intermediate level pieces is 6 and is based on an 11x11 board with the High level pieces at their maximum. The maximum number of Intermediate level pieces is 16 and is based on a 13x13 board with High level pieces at their minimum. The minimum number of High level pieces is 7 and is based on an 11x11 board with the Intermediate level pieces at their maximum. The maximum number of High level pieces is 19 and is based on a 13x13 board with Intermediate level pieces at their minimum.

Table 9-3C Recommended Number of Pieces per Levels per Side @ Start of Game

Board Size Low

Inter-mediate

High Check mate

Level Total

Large Board 4 Rows occupied per side at start of game

14x14 Minimum 14 10 13 1 ---

14x14 Normal 14 28 13 1 56

14x14 Maximum 18 28 27 1 ---

15x15 Minimum 15 12 13 1 ---

15x15 Normal 15 31 13 1 60

15x15 Maximum 21 31 26 1 ---

16x16 Minimum 16 12 15 1 ---

16x16 Normal 16 32 15 1 64

16x16 Maximum 22 32 29 2 ---

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Large Board – Table 9-3C

On a large board with four rows occupied, again at least the first row will need to be Low level pieces. If there are different types of Low level pieces used, then consideration can be given to placing some Low level pieces in the second row, directly protecting the weakest Low level pieces in the first row. The minimum number of pieces is 14 based on the front row of a 14x14 board being occupied with Low level pieces. The maximum number of Low level pieces is based on a 16x16 board with the front row being occupied with Low level pieces and there being Low level pieces in up to one-half of the second row (up to 6) for a total of 22 Low level pieces. The minimum number of Intermediate level pieces is 10 and is based a 14x14 board with the High level pieces at their maximum. The maximum number of Intermediate level pieces is 32 and is based on a 16x16 board with High level pieces at their minimum. The minimum number of High level pieces is 13 and is based on a 14x14 board with Intermediate level pieces at their maximum. The maximum number of High level pieces is 29 and is based on a 16x16 board with Intermediate level pieces at their minimum. On a large sized board with four rows occupied, the maximum number of Checkmate level pieces is two on a 16x16 if there is some scheme to require the capture of two pieces for checkmate.

Relationship to Major and Minor Pieces in Traditional Chess

How do the low, Intermediate, High and Checkmate level pieces of Expanded Chess relate to the major and minor pieces of traditional chess? Low, Intermediate, High and Checkmate level pieces are classified to determine starting positions of a game. Major and minor pieces are defined in terms of capability to end a game. A major piece + King can obtain a checkmate in traditional chess. A minor piece + King can achieve only a stalemate in traditional chess. Identification of major and minor pieces is important because this lets you know if it is worthwhile to continue play during a game. If both players have only a minor piece, then the game is a stalemate and there is no reason to continue playing the game. Conversely, as long as at least one player

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has a major piece, then this player is capable of winning the game. Table 9-4 presents the type of piece classification of the six pieces of traditional chess both ways.

Table 9-4 Type of Piece: Traditional Chess & Expanded Chess

No. Piece Type of Piece

Traditional Chess Type of Piece

Expanded Chess

1. 3 Queen Major High Level

2. 9 Castle Major High Level

3. f Bishop Minor High Level

4. n Knight Minor Intermediate Level

5. 5 King N/A Checkmate Level

6. z Pawn Minor (Promotion) Low Level

This discussion leads to two questions:

1) What are the Expanded Chess major pieces when considering the King as the Checkmate level piece?

2) What are the Expanded Chess major pieces when considering the Emperor as the Checkmate level piece?

We can begin to answer Question 1) by assuming that other pieces with movements that are similar to the Queen or Castle are the best candidates to be major pieces for the King. The Empress will be a major piece because she exceeds the movements/coverage of a Queen and the Fortress, which has all the movements/coverage of a Castle, also will be a major piece. Beyond these certainties, the Chancellor is also a good candidate for a major piece because of his local high density movements/coverage. The Catapult, Ram, Lance and Angel might also

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be good candidates for major pieces because of their unlimited forward movement/coverage. The final answer to Question 1) is currently incomplete and the answer will only be completely provided through the experience of playing and studying a significant amount of Expanded Chess.

With Expanded Chess, the situation becomes even more complicated and interesting because the next two questions that will be asked are:

3) What are the Expanded Chess multi-piece combinations of minor pieces that still result in a stalemate?

4) What are the Expanded Chess multi-piece combinations of minor pieces that still can result in a checkmate?

Again, the answer to Questions 2), 3) and 4) is that they are currently unanswered and that the answers will only be provided through the experience of playing and studying a significant amount of Expanded Chess. Seeking the answer for the multi-piece combinations should be limited to two to four piece maximum. It can be imagined that any multi-piece combinations of a greater number than four would be an overwhelming undertaking because of the huge numbers of interactive combinations of movements still available. This will be one on the interesting mysteries to be solved for Expanded Chess. The challenge has now been issued for any and all to undertake, and the results will take some time to determine.

We have quantified the types and numbers of pieces at the start of the game in this chapter and defined Low, Intermediate, High and Checkmate level pieces. We have also shown how to scale up the number of each type of piece per level and overall as the game board gets progressively larger. Finally we have left the question open as to what are the minor and major pieces in Expanded Chess. And now we will determine the placement of the pieces at the start of the game in the next chapter.

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Chapter 10 – Placement of Pieces at Start of Game

Guidelines

This chapter discusses the approach developed to determine the placement of playing pieces at the start of the game as the size of the game board increases. Parameters to be considered include:

arrangement of the levels

arrangement of the types of pieces

arrangement of the pieces.

Game boards, sized from 9x9 to 16x16, are presented with the recommended locations for the different levels of pieces at the start of the game. The placement of pieces at the start of the game follows basic guidelines so that Expanded Chess will play with the same general feel as traditional chess.

These five major guidelines include:

1) Placing the Low level pieces in the front two row(s) only

2) Placing the Intermediate level pieces in the middle row(s), if applicable (i.e. for 3 or 4 rows occupied per side at the start of the game).

3) Placing the Intermediate level pieces in the outer columns of the back row(s), if applicable.

4) Placing the High level pieces in the center and most outside columns of the back row(s), if applicable.

5) Placing the Checkmate level piece(s) at the back-dead-center position of the game board

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Low Level Pieces

In traditional chess, Pawns are Low level pieces. In Expanded Chess the Knaves and Squires are also Low level types of pieces. These pieces can only move forward and do first fill the front row at the start of the game so that their mobility is maximized. Being in the front row, they cannot be held back from moving forward by their own pieces. If they were placed in the middle or back rows, this possibility exists. If Low level pieces were to start the game in the second row, it should be only after the first row is filled and should be placed on the outside squares of the second row.

Intermediate Level Pieces

In traditional chess, the Knight is the Intermediate level piece. Intermediate level pieces should be placed in the second, third or fourth rows at the start of the game, depending on the size of the game board. Generally, these pieces are placed in the second row, across the width of the row. When there needs to be Intermediate level pieces in the third or fourth rows, these pieces are placed on the outboard portion of the third and fourth rows, especially if the pieces are in the fourth row. The inboard portion of the third and fourth rows is generally reserved for High level and Checkmate level pieces.

High Level Pieces

In traditional Chess the Bishop, Castle and Queen are considered High level pieces. The High level pieces are generally placed in the back row and toward the center of the board. The inboard portion of the third and fourth rows is generally reserved for High level and Checkmate level pieces. The back corners of the board are reserved for High level pieces.

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Checkmate Level Pieces

In traditional chess the King is the Checkmate level piece. In Expanded Chess the Checkmate level piece can be either the King or the Emperor. The placement of the Checkmate level requires some explanation for Expanded Chess. The location of the White King in traditional chess is along the back row square E1 the dark square just to the right of the center of the board on an 8x8 board. The White Queen is located on square D1 which is the light square just to the left of center of an 8x8 board. In Expanded Chess, the Checkmate level piece should always be placed on the back row dead center. There are two options for the Checkmate level piece. On a game board with an odd number of rows wide, the center column of the back row is reserved for the Checkmate level piece at the start of the game. If there is an even number of rows wide, the checkmate piece will occupy the column on the side of center with the dark square. See Table 10-1 for a detailed description of the placement of the King/Emperor and Queen/Empress at the start of the game. This represents a similar approach to what is done in traditional chess.

Table 10-1 Placement of Checkmate Level Piece at the Start of the Game

Size of

Board

Type of

Board

King / Emperor

Location

Queen / Empress

Recommended

Location

Comments

8x8 Even e01 e08 d01 d08

9x9 Odd e01 e09 d01 f01 d08 f08 King @ Center

10x10 Even e01 e10 f01 f10

11x11 Odd f01 f11 e01 e11 g01 g11 Emperor @ Center

12x12 Even g01 g12 f01 f12

13x13 Odd g01 g13 f01 f13 h01 h13 Emperor @ Center

14x14 Even g01 g14 h01 h14

15x15 Odd h01 h15 g01 g15 i01 i15 Emperor @ Center

16x16 Even i01 i16 h01 h16

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Positions of Level of Pieces @ Start of Game

Tables 10-2 through 10-10 present the recommended locations of each of the four levels of pieces at the start of the game for game boards from 8x8 to 16x16 as was discussed in detail in Chapter 9and Tables 9-2 and 9-3A, 3B & 3C. Refer to Table 9-1 for a complete description of the pieces in each level. Table 10-2 represents an 8x8 board and uses the standard FIDE algebraic notation for the location of each square on the game board. The “L1 to L8” in Row 02 and Row 07 indicates that theses rows should be filled with eight (8) Low level pieces, which are Pawns in traditional chess. The “H1 to H5” in Row 01 and Row 08 indicates that theses positions should be filled with High level pieces at the start of the game. In traditional chess these pieces are the Queen, Bishop and Castle.

Table 10-2 8x8 Game Board Positions of Level of Pieces @ Start of Game

L= Low Level I= Intermediate Level H = High Level C = Checkmate Level

08 H1

9 I1

n H2

f H3

3 C1

5 H4

f I2

n H5

9

07 L1

z L2

z L3

z L4

z L5

z L6

z L7

z L8

z

06

05

04

03

02 L1

z L2

z L3

z L4

z L5

z L6

z L7

z L8

z

01 H1

9 I1

n H2

f H3

3 C1

5 H4

f I2

n H5

9

a b c d e f g h

The “I1 and I2” in Row 1 and Row 8 indicate that these positions should be filled with Intermediate level pieces at the start of the game. In traditional chess this piece is the Knight. The position in Row 01 and

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Row 08 filled with “C1” is the Checkmate level and is occupied by the King in traditional chess. As the game board increases in size from 9x9 to 16x16, the positions at the start of the game are filled with Low, Intermediate, High and Checkmate levels as indicated by Tables 10-3 to Table 10-10.

Next when we look at Table 10-3, which has a 9x9 game board, we see that we have extended the conventional grid upward by adding Row 09 and rightward by adding Column h. This concept continues as we make the game board larger up to 16x16 and continue extending rows upward and columns rightward. The bottom left corner square will always be the dark square and always be labeled as a01 for all game boards. Refer to Chapter 7 for complete details of the standard FIDE algebraic notation.

Squares c01, c09, g01 and g09 have two levels of pieces as indicated by the “I3” above the “H5” in square c01. The piece in the upper location in that square is the preferred level of piece for that square. The level for each piece at the start of the game is determined by eliminating one of the two levels in each square (which level is the choice of the game creator) for every square that has two levels shown in Tables 10-3 to 10-10 for that game board size. The level for two symmetrical locations both across the width of the board and over both player sides of the board generally should be the same level and even the same type of piece. As an example, Squares c01, c09, g01 and g09 all contain the same level (either H or I) and even the same type of piece to maintain symmetry.

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Table 10-3 9x9 Game Board Positions of Level of Pieces @ Start of Game

L= Low Level I= Intermediate Level H = High Level C = Checkmate Level

09 H1 I1 I3

H5 H2 C1 H3

I4 H6

I2 H4

08 L1 L2 L3 L4 L5 L6 L7 L9 L9

07

06

05

04

03

02 L1 L2 L3 L4 L5 L6 L7 L9 L9

01 H1 I1 I3

H5 H2 C1 H3

I4 H6

I2 H4

a b c d e f g h i

As the game board expands beyond two rows per side at the start of the game for boards 11x11 or greater in size, the positions of the levels becomes more complicated. Table 10-5 presents an 11x11 board with three rows of pieces at the start of the game for each player. The front row is populated with Low level pieces L1–L11 for each player. The middle row is populated with a combination of Intermediate and High level pieces I1–I6 & I11–I13 with a greater concentration of High level pieces H7-H11 in the centre of the middle row directly in front of the Checkmate level piece C1 (King or Emperor). The back row contains the Checkmate level piece C1 in the center of the row. This piece is flanked by High level pieces H2-H5 on both sides. Outboard of these High level pieces, there are Intermediate level pieces I8-I10 with optional High level pieces H12-H15. Finally on the back corners are two more High level pieces H1&H6 to complete the positioning. The approach is similar for the larger 12x12 and 13x13 three-row game boards.

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Table 10-4 10x10 Game Board Positions of Level of Pieces @ Start of Game

L= Low Level I= Intermediate Level H = High Level C = Checkmate Level

10 H1 I1 I3

H6 H2 C1 H3 H4

I4 H7

I2 H5

09 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10

08

07

06

05

04

03

02 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10

01 H1 I1 I3

H6 H2 C1 H3 H4

I4 H7

I2 H5

a b c d e f g h i j

Table 10-5 11x11 Game Board Positions of Level of Pieces @ Start of Game

L= Low Level I= Intermediate Level H = High Level C = Checkmate Level

11 H1 I9

H12

I7 H14

H2 H3 C1 H4 H5 I8

H15

I10 H13

H6

10 I1 I2 I3 I11 H10

I13 H8

H7 I14 H9

I12 H11

I4 I5 I6

09 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

08

07

06

05

04

03 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

02 I1 I2 I3 I11 H10

I13 H8

H7 I14 H9

I12 H11

I4 I5 I6

01 H1 I9

H12

I7 H14

H2 H3 C1 H4 H5 I8

H15

I10 H13

H6

a b c d e f g h i j k

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Table 10-6 12x12 Game Board Positions of Level of Pieces @ Start of Game

L= Low Level I= Intermediate Level H = High Level C = Checkmate Level

12 H1 I9

H16 I7

H18 H2 H4 C1 H5 H6 H7

I8 H19

I10 H17

H9

11 I1 I2 I3 I11 H14

H10 H11 H12 H13 I12 H15

I4 I5 I6

10 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12

09

08

07

06

05

04

03 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12

02 I1 I2 I3 I11 H14

H10 H11 H12 H13 I12 H15

I4 I5 I6

01 H1 I9

H16 I7

H18 H2 H4 C1 H5 H6 H7

I8 H19

I10 H17

H9

a b c d e f g h i j k l

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10-7 13x13 Game Board Positions of Level of Pieces @ Start of Game

L= Low Level I= Intermediate Level H = High Level C = Checkmate Level

13 H1 I9

H14

I11 H16

H2 H3 H4 C1 H5 H6 H7 I12 H17

I10 H15

H8

12 I1 I2 I3 I7

H18

I3 H12

I15 H10

H9 I16 H11

I14 H13

I8 H19

I4 I5 I6

11 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13

10

09

08

07

06

05

04

03 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13

02 I1 I2 I3 I7

H18

I3 H12

I15 H10

H9 I16 H11

I14 H13

I8 H19

I4 I5 I6

01 H1 I9

H14

I11 H16

H2 H3 H4 C1 H5 H6 H7 I12 H17

I10 H15

H8

a b c d e f g h i j k l m

As the game board expands beyond three rows per side at the start of the game for boards 14x14 or greater in size, the positions of the levels becomes again more complicated. Table 10-8 presents a 14x14 board with four rows of pieces at the start of the game for each player. The front Row 04 is populated with Low level pieces L1–L14 for each player. Row 03 is populated with a combination of optional Low level pieces L15-L18, Intermediate level pieces I1–I6 & I11–I16 and optional High level pieces H18-H19 & H26-H27. Row 02 is populated with a greater concentration of High level pieces H10-H13, optional H14-H15 & H24-H25 in the centre of the directly in front of the Checkmate level piece C1 (King or Emperor), optional H16 & H17 on the outside squares and Intermediate pieces I7-I10, I17–I18, & I25-I26 on the outer portions of the row. The back Row 01 contains the Checkmate level piece C1 in the center of the row. This piece is flanked by High level pieces H2-H8 on

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either side. Outboard of these High level pieces, there are Intermediate level pieces I19-I22 with optional High level pieces H20-H23. Finally on the back corners are two more High level pieces H1 & H9 to complete the positioning. The approach is similar for the larger 15x15 and 16x16 four-row game boards.

Table 10-8 14x14 Game Board Positions of Level of Pieces @ Start of Game

L= Low Level I= Intermediate Level H = High Level C = Checkmate Level

14 H1 I21 H20

I19 H22

H2 H3 H4 C1 H5 H6 H7 H8 I20 H23

I22 H21

H9

13 I25 H16

I7 I8 I17 H24

I27 H14

H10 H11 H12 H13 I28 H15

I18 H25

I9 I10 I26 H17

12 I11 L15

I12 L17

I1 I2 I3 I15 H26

I23 H18

I24 H19

I16 H27

I4 I5 I6 I13 L18

I14 L16

11 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14

10

09

08

07

06

05

04 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14

03 I11 L15

I12 L17

I1 I2 I3 I15 H26

I23 H18

I24 H19

I16 H27

I4 I5 I6 I13 L18

I14 L16

02 I25 H16

I7 I8 I17 H24

I27 H14

H10 H11 H12 H13 I28 H15

I18 H25

I9 I10 I26 H17

01 H1 I21 H20

I19 H22

H2 H3 H4 C1 H5 H6 H7 H8 I20 H23

I22 H21

H9

a b c d e f g h i j k l m n

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Table 10-9 15x15 Game Board Positions of Level of Pieces @ Start of Game

L= Low Level I= Intermediate Level H = High Level C = Checkmate Level

15 H1 H2 I25 H19

I23 H21

H3 H4 H5 C1 H6 H7 H8 I24 H22

126 H20

H9 H10

14 I28 H16

I7 I8 I9 I21 H23

I30 H14

H12 H13 H14 I31 H15

I22 H24

I10 I11 I12 I29 H17

13 I13 L16

I14 L18

I15 L20

I1 I2 I3 I19 H25

I27 H18

I20 H26

I4 I5 I6 I16 L21

I17 L19

I18 L17

12 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15

11

10

09

08

07

06

05

04 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15

03 I13 L16

I14 L18

I15 L20

I1 I2 I3 I19 H25

I27 H18

I20 H26

I4 I5 I6 I16 L21

I17 L19

I18 L17

02 I28 H16

I7 I8 I9 I21 H23

I30 H14

H11 H12 H13 I31 H15

I22 H24

I10 I11 I12 I29 H17

01 H1 H2 I25 H19

I23 H21

H3 H4 H5 C1 H6 H7 H8 I24 H22

126 H20

H9 H10

a b c d e f g h i j k l m n o

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Table 10-10 16x16 Game Board Positions of Level of Pieces @ Start of Game

L= Low Level I= Intermediate Level H = High Level C = Checkmate Level 16 H1 H2

I25 H22

I23 H24

H3 H4 H5 C1 H15C2

H6 H7 H8 I24 H25

I26 H23

H9 H10

15 I29 H18

I7 I8 I9 I22 H26

I31 H16

H11 H12 H13 H14 I32 H17

I23 H27

I10 I11 I12 I30 H19

14 I13 L17

I14 L19

I16 L21

I1 I2 I3 I20 H28

I27 H20

I28 H21

I21 H29

I4 I5 I6 I17 L22

I18 L20

I19 L18

13 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16

12

11

10

09

08

07

06

05

04 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16

03 I13 L17

I14 L19

I16 L21

I1 I2 I3 I20 H28

I27 H20

I28 H21

I21 H29

I4 I5 I6 I17 L22

I18 L20

I19 L18

02 I29 H18

I7 I8 I9 I22 H26

I31 H16

H11 H12 H13 H14 I32 H17

I23 H27

I10 I11 I12 I30 H19

01 H1 H2 I25 H22

I23 H24

H3 H4 H5 C1 H15C2

H6 H7 H8 I24 H25

I26 H23

H9 H10

a b c d e f g h i j k l m n o p

The already presented Tables 10-3 through 10-10 serve as a guide to place actual types of pieces at each position on the game board. Actual games are presented in Chapter 15 following the guidelines developed here. However, before proceeding to actual games developed for Expanded Chess, there needs to be additional work done. We need to review the other major existing features of traditional chess and determine how these features will be incorporated into Expanded Chess. The next chapter begins this determination by reviewing the capture of pieces in general and capture of the checkmate piece.

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Chapter 11 – Penalties and Rewards for Capture and Check

Capture of Playing Pieces

In traditional chess, the capture of an opposing player‟s piece results in the removal of that piece from the board for the duration of the game. Since this a completely deterministic action, it does not expand the serendipitous nature of traditional chess and removes the potential for some chance to be part of the game which is a goal of Expanded Chess. A significant benefit of adding some chance to the game makes it much more difficult for computers to use brute force methods to calculate deterministic outcomes and therefore win games.

In order to expand beyond traditional chess, the capture aspect of the game can be enhanced to implement a richer, more diverse system of rewards and penalties for the capture of an opponent‟s piece. Is this necessary? Yes, Expanded Chess will require enhancement because of the large board size, more pieces in play, greater movement capabilities of the pieces, longer game times and more moves to checkmate. There are two major reasons. These are:

1. Keeping the execution of the game interesting by providing various rewards and penalties to achieve goals during the opening, middle and end games.

2. Providing significant rewards for skillful play and thoughtful risk taking and penalizing poor play and stalling tactics to produce significant material and positional benefits to ideally eliminate stalemates for weaker players because of unconsidered game design.

Let us first define a penalties and rewards for chess. Penalties are actions that reduce the effective strength of the penalized player. Rewards are actions that increase the effective strength of the rewarded player.

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Penalties and Rewards for Capture

Capture in Expanded Chess can have the possible consequences as shown in Table 11-1.

Table 11-1 Possible Penalties or Rewards of Capture for Expanded Chess

Capturing Player

Player whose Piece has been Captured

Penalties (reduces

effectiveness)

N/A Remove captured piece from the game board for the duration of the game

N/A Move captured piece to a new location chosen by capturing player

Rewards (increases

effectiveness)

Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

N/A

Move capturing piece to a new location and leave captured piece on board in present location

N/A

The question resulting from Table 11-1 becomes: what combination of penalties and rewards should be imposed for a capture? Now let us review the advantages and disadvantages and implementation of each of the aforementioned possible penalties and rewards. Removing a captured piece from the game board for the duration of the game is the penalty used in traditional chess. This is a simple, effective penalty that helps drive the game to completion and weakens the player who is not highly skilled at defending his own pieces. This penalty will also generally be used in Expanded Chess as a standard rule. Moving a captured piece to a new location chosen by the capturing player is another possible penalty for Expanded Chess. When would it make sense to move a captured piece to a new location rather than remove it from the game board? The situation where this might be a penalty would be to capture an advanced Pawn and place it back in

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front of a Castle to impede access of the Castle to the open board. A Pawn or other low level piece might also be placed near its checkmate piece with the intent of blocking escape or reducing necessary coverage to secure a checkmate. This rule will be incorporated into future generations of Expanded Chess as playing experience illustrates best how to implement this rule. Currently, there is no specific reward for the capturing player. One possible reward includes the return of a player‟s previously captured piece of equal or lesser strength to the game board. This reward could be implemented, as an option, in lieu of removal of the captured piece. Although this option might appear to be attractive, the challenge for implementation is where to re-locate the capturing piece. The piece being captured needs to maintain its position for the move. The solution to this challenge is not to re-locate the capturing piece and hold its pre-capture location. The location of placement of the returned piece would be such that it was not directly able to place opponent‟s pieces at risk of capture or check with its next move. The piece being captured is permitted to move out of the capture location after the returning piece is placed on the board as the move for the capturing player. This rule will be also incorporated into future generations of Expanded Chess as playing experience illustrates best how to implement this rule.

Moving the capturing piece to a new location and leaving the captured piece on the board in its present location is also a possible reward for capture. The move would be optional (in lieu of removing the captured piece) and the move would be a move that is normally possible from the location of the capture. A scenario where this might be attractive would be for example, a Queen foregoing a Pawn capture and removal and using the optional move continue to a square to place the opponent‟s King in check or capture a higher value piece. Instituting a rule of this type has the potential to open up a huge combinoric can of worms and might throw the defensive concepts of guarding and trading into turmoil if a piece could make a cascade move similar to checker except for not removing the pieces. Therefore, this move will be limited to foregoing one capture and the piece foregoing capture on the first leg of

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such a move will only be allowed to play the second leg of the move as a normal movement and will not be allowed forego capturing a second piece.

Penalties and Rewards for Check

As the size of the game board and the complexity of the game increase, more opportunities will be needed to move Expanded Chess towards Checkmate, especially during endgame play. One of the more annoying weaknesses of traditional chess is the situation where a player can be placed in check many times in sequence but cannot be mated because the offensive player does not inherently have enough firepower to defeat the opponent. One way to simple and effective way to increase a player‟s firepower and decrease an opponent‟s resistance exactly at the right time that it is required and deserved is for the game to provide penalties and rewards when a player is placed in check. Once a player place the opponent in check several times in sequence, there needs to be game features that presents a strong opportunity for the player to checkmate the opponent so that the game does not drag on for 200 or 300 moves. Table 11-2 presents possible penalties and rewards for placing an opponent in check for Expanded Chess.

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Table 11-2 Possible Penalties or Rewards of Check for Expanded Chess

Checking Player who has placed an Opponent’s Piece in Check three or more times

or in Sequence

Player whose Piece has been Checked three or

more times in Sequence

Penalties (reduces

effectiveness) N/A

Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

N/A

Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Rewards (increases

effectiveness)

Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

N/A

Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

N/A

The rewards and penalties in Table 11-2 will be applicable once for each time a player places the opponent in check three or more times in sequence. The advantages of the rewards and penalties presented in Table 11-2 are self-evident. Implementation of these rewards and

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penalties will reduce the possibility of games going into check stall, reward a player who attacks aggressively, maintain game intensity and drive games to more certain outcomes with reasonable move counts. The only disadvantage for adding these features is making Expanded Chess more slightly complicated and rules driven. However, the large advantages already discussed far outweigh small disadvantage of including new penalties and rewards.

Penalties and Rewards Summary

Ideally, each game of Expanded Chess should have at least one penalty for being captured, one reward for capturing, one penalty for being checked in sequence, and one reward for checking in sequence, especially for the games played on the larger game boards. These features are intended to speed up game play, increase material exchange activity, and accentuate differences in skillful execution, reward good play with significant opportunities for checkmate and eliminate undeserved stalemates because of inherent weakness in the design of the game features.

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Chapter 12 – Promotion of Playing Pieces

Pieces That Can Be Promoted

Promotion is limited to pieces that can only move in the forward direction. The object of limiting these pieces to forward motion is to ensure the start of the game move forward and generates irreversibility. You do not want these pieces to get involved in an oscillating series of timid attacks and retreats at the start of the game. This could result in significantly slowing down the forward progress of the start of the game. These front row pieces are predestined cannon fodder and need to take one for the team to keep the game moving ahead.

The object of promotion is to reward the skill of a player who is able to advance a forward-only-moving piece deep into the opposition region of the game board without being captured. It is also necessary because once forward-only-moving pieces have reached the far end of the board. They are essentially rendered immobile and useless unless they can move in another direction. At this point a player would be penalized for succeeding because the only reward for reaching the far end of the board is being frozen. Therefore, promotion of the piece is necessary for that piece to regain mobility and utility. These three pieces are limited to forward only motion and therefore will require promotion. These pieces are:

Squire

Knave

Pawn

Range and Value of Promotable Pieces

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Range of a Promotable Piece

In theory, the value of a forward-only moving piece does not reflect the fact that as the piece advances its range is reduced and its value decreases. It also does not reflect that the piece only has a maximum number of moves which is less than the number of rows deep of the game board. In practical terms, this limitation is not very significant because the number of moves in a game is finite and relatively short. Therefore, most pieces have only a small portion of the total moves for one side for the game. So in practical terms, although non-promotable pieces have a small and fixed number on moves, it does not significantly impact their value because in a real game those moves will not all be consumed because either the pieces will be captured or the game will be completed before all their available moves are consumed. If most of the Low level pieces were stalled at the far end of the board at the end of the game, then it could be argued that the Low level pieces are inherently undervalued and not being properly utilized in the overall scheme of the game. Unless this situation becomes a common situation in Expanded Chess, it can now be assumed that promotable pieces are not under-valued to any great extent.

Value of a Promotable Piece

This is a somewhat arbitrary exercise for two reasons, which are:

(1) these pieces have a restricted value because of their ability to move forward only and have a limited number of moves

(2) these pieces are promotable; therefore they have some limited probability of becoming a higher value piece and therefore increasing in overall value some small but intangible amount.

These two factors should tend to offset each other and make the exercise of accounting for these factors not truly significant. Currently they are not accounted for in the value calculation for Expanded Chess. As

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experience is gained from game play the subject can be re-visited in the future.

Quantifying Promotability

In Expanded Chess, there are two possible approaches to award promotion to a piece. Either one or some combination of both of these two approaches can be used in Expanded Chess.

The first approach is similar to traditional chess is called Direction Promotion. In traditional chess any piece is available for promotion but the Queen is the piece which is promoted to almost 100 % of the time because it is the most powerful piece. Since Expanded Chess can have many more types of pieces at the start of the game, and generally only the most powerful pieces are used in a promotion, it then makes sense to limit promotion to the most powerful pieces. To achieve some variety and flexibility, Expanded Chess will allow promotion to the four most powerful types of pieces that can be used for promotion. This is the most simple and effective way to approach promotion and follows the approach of traditional chess.

Since Expanded Chess games will require more moves for completion and have more promotable pieces, which themselves are more mobile and stronger than Pawns, therefore more promotions and promotions earlier in the game might occur. The second approach to promotion accounts for this potentially increased role for promotion. The concept is to establish a promotability reward factor and reward skillful play by giving the pieces with the least chance of being promoted the greatest reward for actually achieving promotion. In simple terms, the promotion of a Pawn will entitle the player to a piece of greater strength than promotion of a Knave or Squire because the Pawn is weaker and less mobile and if it achieves promotion, it should be entitled to a greater reward.

The Promotion Reward Factor is based on the following criteria:

Strength (value) of the piece being promoted

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number of pieces of the type on the board

size of the game board

Each one of these criteria is assigned a numerical value based on the relative strength of the piece and the criteria for that piece are summed together to determine a final reward factor. Each criterion is discussed in some detail following.

Effect of Strength (value) of Piece Being Promoted

A piece of lesser value will be rewarded with a greater range of pieces which it can be promoted. As an example, a Pawn which has a value of 1.0 on all game boards can be promoted to the highest value pieces

Promotability is inversely proportional to the value of the piece being promoted because the greater the value of the piece, the greater the chance it can achieve promotion because it is inherently has better mobility to defend itself and is less likely to be sacrificed or exchanged. Therefore a higher value piece should be less rewarded when it is promoted.

No. of Pieces of Specific Type at Start of Game

The more pieces of one specific type there are on the board at the start of the game, the greater the chance that one of this type can achieve promotion. Therefore the piece should be less rewarded if it is promoted.

Size of the Game Board

The larger the game board, the greater the probability that a piece will not survive to reach the opposite end of the board and achieve promotion. This is especially for the number of rows deep that the board is, if a board is not square. As an example, it is more probable that for a board of 80 squares, it is more difficult to achieve promotion on a board that is 10 rows deep x 8 columns wide than on a board that is 8 rows

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deep x 10 columns wide. The deeper board requires the piece to advance two more rows to achieve promotion.

Calculating Promotion Reward Factor

Promotability Reward Factor is determined to evaluate numerically how much of a reward is earned for promotion in a given circumstance.

The basic formula is Promotion Reward Factor (PRF) is:

PRF = (D1 x Rd x Cw x Bp) / (Np x Vp) [Eqn. 12-1]

where:

D1 = Depth coefficient = Rd/Cw

Rd = Rows deep of the board

Cw = Columns wide of the board

Vp = Value of the Piece as assigned in Chapter 5, Table 5-3A & B

Np = No. of pieces of the type being determined

Bp = Base Promotability

Let‟s start with a special case to understand Bp Base Promotability which for a Pawn on a traditional 8x8 chess board is:

Bp0= (D1 x Rd x Cw) / (Np x Vp) where D1 = Rd / Cw

Bp0= (8/8 x 8 x 8) /(8 x 1)

Bp0 = 8

This equation tells us that a Pawn should be rewarded in promotion with a piece that is up to 8 times the value of the Pawn. This corresponds well with traditional chess where the value of the Queen is 9 times the value of a Pawn. To obtain the base promotability we make the adjustment for the predicted reward factor to the actual reward factor:

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Bp = 9 / Bp0

Bp = 9/8 =1.125

This formula gives us a workable, sensible quantitative approach to rewarding promotion.

Next we determine the Promotion Reward Factor (PRF) for a new Expanded Chess piece. Let‟s determine the PRF for a Knave on a 12 Rd x 10 Cw board.

PRF = (10/12 x 12 x10 x 1.125) / (12 x 1.51)

PRF = 6.209

This calculation tells us that a Knave that achieves promotion should be rewarded with a piece with a value of up to 5.472 under these circumstances.

Table 12-1 summarizes the calculation for all the promotable pieces for the range of game board sizes. Figure 12-1 plots the values in Table 12-1.

Table 12-1 Promotion Reward Factors

Columns wide

Squire Knave Pawn

9 4.010 6.002 9.063

10 4.485 6.743 10.181

11 5.001 7.484 11.301

12 5.472 8.226 12.422

13 5.966 8.969 13.543

14 6.018 9.047 13.661

15 6.509 9.785 14.775

16 7.000 10.524 15.891

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Promotion Reward Factors

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

9 10 11 12 13 14 15 16

Columns Wide Board Size

Pro

mo

tio

n F

ac

tor

Squire

Knave

Pawn

Figure 12-1

Table 12-2 summarizes the results of the Promotion Reward Factor calculation and uses the 12 columns wide board as an estimated average for selection of the promoted pieces by closely matching the PRF value with the Promoted Piece value based on the a 12 columns wide board. Refer to Table 12-1 for the value of the PRF for the promotable pieces on different size boards. The PRF values match well with the Promoted Piece values except for the Pawn. The Pawn PRF is somewhat less than the value of the Fortress because the Fortress was selected over the closer match of the Emperor which has a value of 12.87. The Emperor was not selected because it is normally the checkmate piece and is not acceptable as a Promoted Piece because one side is not allowed to have two pieces which are available for checkmate.

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Table 12-2 Promotion Reward - Maximum Value Piece Available for Promotion

Piece Being Promoted Squire Knave Pawn Average value PRF for 12 Columns wide

Board 5.472 8.226 12.422

Promoted Piece (Value on 12 Columns wide Board)

Armory (5.60)

Duke (8.67)

Fortress (14.45)

Combining the Two Approaches for Promotion

As discussed previously in the chapter, Expanded Chess will combine the two approaches, Direct Promotion and Promotion Rewards Factors, to achieve promotion for each Expanded Chess game. There are three possible ways to combine the two approaches. Table 12-3 presents the combinations for Expanded Chess.

Table 12-3 Combining Approaches for Promotion

Approach

Type

Direct Promotion

Combination of Approaches Promotion

Reward Factor

1 Yes, entire

game None None

2 Yes

Yes, allow direct promotion a limited number of times with the player choosing when these promotions will be used. All other promotions are by using Promotion Reward Factor.

Yes

3 None None Yes, entire

game

Approach Type 1 is the approach used in traditional chess and is familiar to all players of traditional chess. Approach Type 3 takes the opposite tack of Approach Type 1 and uses only the Promotion Reward Factor to achieve promotion during an Expanded Chess game. Approaches Types 2 uses a combination of approaches to achieve promotion. Approach Type 2 allows direct promotion only a specified

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number of times a game with the player choosing when it will be used. Direct Promotion is to be used a fixed number of times per game, generally three times. The other stipulation will be the sequence of the use of direct promotion. Direct Promotion when the player achieving promotion decides. The Promotion Reward Factor will be used for the duration of the game.

Approach Type 2, although somewhat more complicated to implement, accomplishes two purposes of value for overall game quality. The first purpose is again to reduce determinism (but not increase frustrating random chance) in the game by allowing players to choose between two different options, which again helps to reduce an advantage for computers because there are more computing requirements each time an outcome is not predetermined and options require evaluation. The choice for the player is to decide to use direct promotion earlier in the game when more material still populates the game board and gain a relatively lesser overall advantage but obtains the direct promotion for certain or wait until later in the game when adding a powerful piece might be sufficient to gain a relatively larger material advantage and even be enough to secure checkmate but also risking not being presented with the option due to lack of opportunity. The second purpose of Approach Type 2 is reduce the possibility of promotion becoming an potential overkill scenario allowing one player or computer an excessive reward by directly promoting too many pieces and promotion becomes a vehicle achieve more and more promotion until it becomes excessive.

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Chapter 13 – Special Rules for Expanded Chess

Castling

Present day Castling originated in the sixteenth century some time after the Queen and the Bishop were introduced and after earlier versions of Castling were tried. Castling achieves two contradictory goals relatively effectively at one time. These goals include:

placing one Castle in a more effective location to speed up the attack portion of the game

placing the King in a more protected location to improve one‟s defensive standing

These goals are contradictory because the freeing up a Castle to go on the attack does not improve your defensive strength without the Castling move. Attempting to free a Castle early in a game comes with a significant defensive penalty. Freeing a Castle results in unattractive Pawn formations and/or clearing the back row to break out the Castle. This is an inherent weakness in the structure of traditional chess.

Once the game board is larger than two rows occupied at the start of game (an 11x11 or greater sized game board), Castling should be eliminated. Castling is not required or desirable on a large board because:

the Emperor has greater mobility than the King therefore needs less protection

there are too many pieces in between the Emperor and the Castle (or other piece) to move on the larger board

the Emperor has greater defensive resources than a King including more pieces with high strength values on the larger board

the game lacks symmetry when the King is “hiding in the corner”

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it reduces simultaneous attack/defend scenarios and game complexity

So as the game board becomes larger and there are more pieces on the board, Castling becomes an unnecessary move. When there is a larger game board, there are more pieces on the board and more pieces with substantial mobility, so there is no problem getting the attack portion of the game up to speed. Once there are three rows occupied at the start of the game, the checkmate piece is well protected from the opposition for a major portion of the game so there is no longer a need for Castling. This elimination of Castling maintains a higher level of combinoric complexity for Expanded Chess, which makes any computer version of the game less to take excessive advantage of its high speed computing methods by hiding the King and wearing down the opponent by being able to project further into the a simplified game using Castling. This helps level the playing field between computers and humans. The human mind is relatively good at perceiving complex patterns in space and time and exploiting advantages from the perception of these patterns. Eliminating Castling will allow these complex patterns to exist to a greater extent than if Castling were present.

En Passant

To begin any discussion of En Passant, first we need to understand how this somewhat obscure rule is actually implemented. We also need to understand why this rule exists. I believe forces that came into play to create En Passant were:

making the game faster with the two-square opening moves for Pawns

and preserving a higher level of Pawn exchange which would have been occurring in the older, one-square-only opening move for Pawns.

The two-square opening move for Pawns eliminated some of the direct contact between opposing Pawns because a Pawn on its fifth row was

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not able to fork either opposing Pawn on their second row moving to their third row. These Pawns could simply move up two rows and end up directly beside the opposing Pawn in its own fifth row.

When viewed in a historical context, En Passant makes sense. The general goal of the Renaissance European variant of chess was to make the game play faster and promote greater levels of exchange to ultimately produce a winner in a more timely manner. The two-square opening move rule for Pawns assisted in speeding up the game. However, it was also realized that this rule reduced the potential for Pawn exchanges, which also keeps the game moving at a faster pace. Therefore, En Passant was the solution to recover some of this potential for Pawn exchange, when adopting the two-square opening move for Pawns. The question remains: do we need En Passant today?

En passant always seemed to be an arbitrary, anti-intuitive rule that only confuses and frustrates the average chess player. After reviewing the history and intent of the rule, there appears to be some justification for the rule, but I am still not convinced that En Passant needs to exist, generally, in Expanded Chess.

Experience has shown that the two-square opening move for Pawns is effective at speeding up the game and the reduction of early game Pawn exchanges is not a too great price to pay for this benefit. Since the pieces behind the Pawns (such as Queens, Castles and Bishops) have extended movement capability, high amounts of early Pawn exchanges would lead to potential overexposure where opposing, extended-range pieces, such as Queens, are directly exposed to each other and might end up being exchanged very early in a match. With the older Arab pieces having a limited range of movement (the firz and fil in Shatranj), early game Pawn exchanges would not have had the potential to impact the Shatranj game as dramatically because the of the limited range of these weak pieces. The exchange of Pawns would not immediately put these weak pieces, which are behind the exchanged Pawns, in direct contact with their opposing counterparts. Therefore, it can be justified that once the history and intent of En Passant is understood, that En Passant can

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be eliminated from Expanded Chess because its elimination will not noticeably impact Expanded Chess.

Expanded Chess has Low level pieces (Squire and Knave) which can move multiple squares forward each move (2 Squire, 3 Knave) to keep the game moving at a quick pace. Since there are even more High level pieces with extended range in Expanded Chess, early exchanges of Low level pieces have significant potential to impact the game. In terms of game design strategy, reducing the potential for exchanging High level pieces very early should be more fair for human against computers because maintaining a busy game board gives the computer less time to explore a board with many possible moves in depth.

So once game board is larger than two rows clear at the start of game, (for boards 11x11 or larger) Expanded Chess will eliminate En Passant. Since 9x9 and 10x10 games will be quite similar to traditional chess, En Passant can be preserved for the sake of continuity with traditional chess.

Replacement of a King on a Larger Board Game

On a larger game board, consideration should be given to replacement of the King with the Emperor. This is especially true when the Emperor is selected as a playing piece. Consider 10x10 the minimum board size for replacing the King with an Emperor. Also include replacing the Castles with Fortresses on the back row corners if you want to begin upgrading traditional chess in a substantial way. This larger board gives the emperor some room to move about as compared to a King.

What is the best size of board to consider replacing the King? Probably 11x11 or larger. Why this size? Once three rows are occupied at the start of the game and castling is removed from game, the King should be replaced with the Emperor. The Emperor is a more powerful and mobile piece than the King and is needed to balance the game as the game

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board grows larger and the game pieces become stronger and more numerous.

Multiple Players / Multiple Movements per Turn

When the game board becomes very large and there are many pieces on the board, it becomes an important consideration to keep the game moving at a crisp pace and not allow the game to get so bogged down that it takes 200 moves to Checkmate. One technique to keep the game interesting and fast paced is to give each player multiple moves for each turn during the game. There are some different possibilities for incorporating multiple moves into Expanded Chess. These include having two, three or four moves per turn for each side and including multiple players per side to exercise these moves.

Refer to Chapter 8, Table 8-1 and Table 8-2 for a discussion of the number of players per side and the number of moves per side as the size of the game board increases. With larger games, more than one player per side can participate and multiple movements per turn can be shared by teammates. This team approach can be a great way to help new and younger players learn and develop their Expanded Chess skills.

Slide Bishop

The Slide Bishop is needed on game boards which are an odd number of rows wide. On this type of game board, bishops who are placed symmetrically about the center column both end up being placed on the same color square. If the bishop maintains his established movements, then both Bishops are limited to the same color squares. This restriction significantly reduces the effectiveness of the Bishops, especially near the end of the game if both Bishops are only able to move along one color of diagonals. If the King /Emperor

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stays on the opposite color squares, then the Bishops cannot provide direct checking capability, which limits their effectiveness. One solution to this problem is to enhance the movement of the Bishop by allowing it to slide one square sideward to the left or right. This allows the Bishop to “slide” to a square of the opposite color of the one he was originally placed. This slight modification to the Bishop‟s movement restores and enhances his capability and effectiveness. There are three alternatives solutions to solve this problem. The first alternative solution is not to use the Bishops on odd column width game boards and avoid the situation completely. The second alternative solution is to use more than two Bishops. Three Bishops can be used in an “unbalanced” symmetrical placement arrangement. This arrangement places one Bishop on one color (assume light) and two Bishops on the opposite color (assume dark color). The final possible arrangement is to have four Bishops and have two on light squares and two on dark squares. When two are placed on each side of center symmetrically, this achieves a symmetrical configuration without requiring the slide characteristic. This final alternative is the preferred solution for larger game boards (11x11 or greater) where there are three rows occupied by each side at the start of the game and there is enough space to make this option practical.

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Chapter 14 – Describing and Cataloging New Games

Major Game Parameters

Since there will be a vast array of combinations of new chess games when the guidelines of Expanded Chess are applied, it will be necessary to create a system to catalog each new version of Expanded Chess. It will be necessary to tag the major parameters of any new game to describe the game properly. The major parameters include:

the dimensions of the game board

the number of rows occupied at the start of the game

the number of players per side

the pieces each player can move

the number of moves in turn for each side

the total number of pieces per side

the number of types of pieces

the location of each piece at the start of the game

the number of each type of piece

the piece being checkmated

the pieces available for promotion

the arrangement of the pieces on the game board

the rewards and penalties of capture

the rewards and penalties of check

special rules

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In order to define a new game in Expanded Chess, Tables 14-1A and Table 14-1B are needed to provide a game description that will be completed with the development of each new game in Expanded Chess. Table 14-1A is completed with data for traditional chess as an example. The requirements for each item described are listed in the right hand section of the table. Table14-1B is completed by placing each piece in its proper location on the game board at the start of the game. All games start from the bottom left corner in square a01 with the white side along the bottom rows of the board. The black side is placed in the appropriate location in the upper board based on the size of the game board regardless of the 16x16 size of the game board used in Table 14-1B. The 16x16 board size is used to envelope the entire range of board sizes available in Expanded Chess (there are 26, refer to Table 7-1) with the use of one full-sized, 16x16 board. Table 14-1B is also completed for traditional chess. Appendix E provides a completed set of game description and game arrangement sheets for the all the new Expanded Chess games described in Chapter 15.

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Game Sheet 1 - Expanded Chess

0123456789abcdefghijklmnopqrstuvwxyz EXPANDED CHESS Game Description

General Items

1 Game Name Traditional FIDE Chess

2 Game Identification Number 0808-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 8

2 Dimensions of the game board – columns deep 8

3 Number of rows occupied at the start of the game per side 2

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 5

2 Total number of types of High level pieces 3 39f

3 Total number of types of Intermediate level pieces 1 n

4 Total number of types of Low level pieces 1 z

5 Total number of types of pieces per side 6

6 Total number of Checkmate level pieces 1 5

7 Total number of High level pieces 5 3 99 ff

8 Total number of Intermediate level pieces 2 nn

9 Total number of Low level pieces 8 zzzzzzzz

10 Total number of pieces per side 16

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 1

3 Multiple moves with any piece Not Included

4 Multiple moves with single piece Not Included

5 Multiple pieces with single move each Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Included

2 En Passant Included

3 Castling Included

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Game Sheet 1 - Expanded Chess

0123456789abcdefghijklmnopqrstuvwxyz EXPANDED CHESS Game Description

General Items

1 Game Name Traditional FIDE Chess

2 Game Identification Number 0808-001

4 Slide Bishop Not Included

1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote z

2 Pieces available for promotion

2a Direct Promotion 39f n

2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4 Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game e01 e08

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Game Sheet 1 Expanded Chess Game Arrangement Game Name Traditional FIDE Chess

Game Identification Number 0808-001

16x16 Game Board

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11

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08 9 n f 3 5 f n 9

07 z z z z z z z z

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02 z z z z z z z z

01 9 n f 3 5 f n 9

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Chapter 15 – Conclusions and New Expanded Chess Games

New Pieces

The purpose of this manual is to expand traditional chess into the rich new multi-dimensional set of games that takes chess into the new millennium and beyond. This manual has presented an industrial strength, rigorous, intellectual design guide to take modern technology head-on, prosper in the world of the Supercomputer and the Internet and keep the game relevant.

The traditional game risks becoming stale and unappealing. Its complexity should be increased to meet the challenges of the computerized world. Expanded Chess has accepted that challenge and presented a rigorous analysis and thoughtful expansion of traditional chess in a way that is intuitive and follows directly from the traditional game. This approach has many benefits. The most obvious benefit is that you do not loose the huge base of traditional chess players. This is accomplished by expanding from traditional chess and following the same philosophies, but expanding the approach of these philosophies in ways that are logical, meaningful and intuitive to traditional chess player.

A part of the history of chess and its variations is one of arbitrary developments and a lack of standardization. The arbitrary nature of these developments has left most of the chess variations with a limited player base. The usual approach to develop a new variation of chess has been to add a couple of new pieces and perhaps expand the board by a row or column or two. There are literally hundreds of expanded variations of chess including an entire web site dedicated to expanded variations of chess. The major issue that I have with most of these variations of chess is that you cannot use what you learned in one expanded variation to play another expanded variation of chess. You have to keep learning the new moves over and over and new pieces each time you learned a new expanded variation of chess. This lack of standardization is the biggest drawback for any expanded variation of chess.

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Expanded Chess has proposed a large base of 36 standard pieces, each with a standard move as shown in Table 3-1 and Appendix A. The 36 pieces gives enough variation of movement that there is no need to tinker with the moves of a new piece in a different variation of Expanded Chess. The main objective of having so many new pieces is to capture and standardize, now and for the future, the concept that each piece will have the same movements and rules governing its play for all the different variations of chess. The game designer simply selects a different standard piece if a new type of movement is needed. There is a major benefit to this approach because many variations of the game can be developed with the same piece base. Players can move intuitively from one expanded game to another because they already know new pieces and their moves.

New Boards

Expanded Chess will have a range of game board sizes. Game boards can vary from 6x6 to 16x16. There are preferred board dimensions especially square boards. However, square boards can have an odd or even number of squares. Boards can also have odd by even dimensions.

The most preferred board sizes include 8x8, 9x9, 9x8, 10x10, 10x9, 11x11, 11x10, 12x12, 12x11, 13x13, 13x12, 14x14, 14x12, 15x15, 15x13, 16x16 and 16x14. Refer to Table 7-1 for a complete listing of the game board sizes.

New Rules

Table 14-1A summarizes the new rules for Expanded Chess. The table is divided into eight sections and summarizes all the options for the new rules for Expanded Chess. Table 14-1 needs to be completed for each Expanded Chess game before playing the new game.

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Final Challenge – The New Expanded Chess Games

The final challenge for is now to get you to play Expanded Chess. The enclosed website download key provides you with a choice of eight new Expanded Chess games to play. The table gives a summary description of each game.

Table 15-1 New Expanded Chess Games

Game No. Size

(RxC)

No. of Pieces per

side L/I/H/C

No. of Pieces per side

Total

No. of Players per side

Checkmate Piece (King

Emperor)

Rows occupied @ start of game per side

Name

0808-001 8x8 8/2/5/1 16 2 King 2 Traditional FIDE Chess

0909-001 9x9 9/2/6/1 18 2 King 2 Perfect Chess

1010-001 10x10 10/2/7/1 20 2 King 2 Breakout Chess

1111-001 11x11 11/6/15/1 33 2 Emperor 3 Next Level Chess

1212-001 12x12 12/12/11/1 36 2 Emperor 3 Arrival Chess

1313-001 13x13 13/16/9/1 39 2 Emperor 3 Lucky 13 Chess

1414-001 14x14 14/28/13/1 56 2 Emperor 4 Steinitz Chess

1515-001 15x15 15/31/13/1 60 2 Emperor 4 Laskers Chess

1616-001 16x16 16/32/15/1 64 2 Emperor 4 Tarrasch Chess

Appendix C provides a completed set of game description and game arrangement sheets for the all the new Expanded Chess games described in Table 15-1. Now go play Expanded Chess and lead the chess world into the new millennium.

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Bibliography

Bibliography (1997) Deep Blue, IBM Research. Available at: http://www.research.ibm.com/deepblue/meet/html/d.3.html (Accessed: 26 December 2008). (2006) Kramnik vs Deep Fritz: Computer wins match by 4:2 , Chessbase News . Available at: http://www.chessbase.com/newsdetail.asp?newsid=3524 (Accessed: 22 December 2008). (2008) Ancient History, Wikipedia. Available at: http://en.wikipedia.org/wiki/Ancient_world (Accessed: 22 December 2008). (2008) Chessopedia – En Passant , Chess.com . Available at: http://www.chess.com/chessopedia/view/en-passant (Accessed: 28 December 2008). (2008) En Passant, Wikipedia. Available at: En Passant (Accessed: 28 December 2008). (2008) Three-dimensional Chess, Wikipedia. Available at: http://en.wikipedia.org/wiki/Three-dimensional_chess (Accessed: 27 December 2008). (no date) Evaluation of Material Imbalances, Dan Heisman. Available at: http://mywebpages.comcast.net/danheisman/Articles/evaluation_of_material_imbalance.htm] (Accessed: 27 December 2008). (no date) What are thepieces worth?, Gothic Chess. Available at: http:/www.gothicchess.com/piece_values.html (Accessed: 27 December 2008). Alburt, L. & Parr, L. (2003) 'Play Chess in One Hour, The Concept of Winning Material', Secrets of the Russian Chess Masters, Fundamentals of the Game, Volume I. New York: W. W. Norton & Company, p. 30. Betza, R. (1996) About the Value of Chess Pieces, The Chess Variant Pages. Available at: http://www.chessvariants.com/d.betza/pieceval/index.html (Accessed: 27 December 2008). Bisguier, A. (no date) Ten Tips to Winning Chess by International Grandmaster Arthur Bisguier, Chess.com. Available at: http://www.chess.com/article/view/ten-tips-to-

winning-chess-by-international-grandmaster-arthur-bisguier (Accessed: 26 December 2008). Davidson, H. (1949) A Short History of Chess, New York : Greenberg .

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Dvorsky, G. (2006) The Future of Chess, Sentiment Developments, Institute for Ethics and Emerging Technologies. Available at: http://ieet.org/index.php/IEET/more/dvorsky20061208/ (Accessed: 22 December 2008). Howe, D. & Bodlaender, H.L. (no date) The Chess Variant Pages. Available at: http://www.chessvariants.com/ (Accessed: 26 December 2008). Hsu, F.H. (2002) Behind Deep Blue: Building the Computer that Defeated the World Chess Champion. 3rd edn. Princeton : Princeton University Press. Hsu, F.H., Campbell , M. & Hoane Jr., A.J. (2001) Deep Blue. Available at: http://sjeng.org/ftp/deepblue.pdf (Accessed: 26 December 2008). Kaufman, L. (1999) 'The Evaluation of Material Imbalances', Chess Life, March, 1999 Keats, V. (no date) Book Extracts, Chess: It Origins Vol. II. Available at: http://www.historyofchess.co.uk/vol2.htm (Accessed: 22 December 2008). Keats, V. (no date) Book Extracts, Chess: It Origins Vol. II. Available at: http://www.historyofchess.co.uk/vol2_2.htm (Accessed: 28 December 2008). Knightly, J. (2006) Chess History, Chess Circle - The Place For Chess. Available at: http://www.chesscircle.net/chess-history/chess-history.html (Accessed: 28 December 2008). Kurzweil, R. (2002) Deep Fritz Draws: Are Humans Getting Smarter, or Are Computers Getting Stupider? . Available at: http://www.kurzweilai.net/meme/frame.html?main=/articles/art0527.html? (Accessed: 22 December 2008). Merriam-Webster Online Dictionary (2009). Available at: http://www.merriam-

webster.com/dictionary/meridian (Accessed: 19 January 2009). Pandolfini, B. (1992) 'Material Evidence', Pandolfini's Chess Complete: The Most Comprehensive Guide to the Game, from History to Strategy . New York: Simon and Schuster, pp. 54-55. Reinfeld, F. (1990) 'Relative Value of the Chess Forces', The Complete Chess Course. New York: Random House, p. 32. Schulz, A. (2006) 'VLADIMIR KRAMNIK TAKES ON 'DEEP FRITZ', The Last Man-Versus-Machine Match?', Spiegel Online International, English, November 24, 2006 [Online]. Available at: http://www.spiegel.de/international/0,1518,450566,00.html (Accessed: 26 December 2008).

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The Tribune, Online Edition, Chandigarh, India (2001) 'Trends and Pointers, Chess Champ Challenges Computer', August 13, 2001 [Online]. Available at: http://www.tribuneindia.com/2001/20010813/edit.htm - 5 (Accessed: 22 December 2008). Trice, E. (2004) '80 Square Chess', ICGA Journal, June, pp. 87-90.

EXPANDED CHESS Appendix A

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Expanded Chess Appendix A

DESCRIPTION OF PIECES &

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Ruling Class Empress – (EPS) The Empress has the combined moves of a Castle, a Bishop and a Guard and is the most powerful piece in Expanded Chess. Her index is 6 6.

Movement/Capture Summary

0 Orthogonal Moves (Castle-like) Forward Unlimited Backward Unlimited Sideward Unlimited

Diagonal Moves (Bishop-like) Forward Unlimited Backward Unlimited L-Type Moves (Knight-like) Forward 3,2 L Backward 3,2 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Ruling Class

Chancellor – (CHL) The Chancellor has the combined moves of an Armory, a Priest and an Archer. His index is 6 5.

Movement/Capture Summary

2 Orthogonal Moves (Castle-like) Forward 3 Squares Backward 3 Squares Sideward 3 Squares

Diagonal Moves (Bishop-like) Forward 3 Squares Backward 3 Squares L-Type Moves (Knight-like) Forward 3,1 L Backward 3,1 L

MOVEMENT/CAPTURE DIAGRAM

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07

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DESCRIPTION OF PIECE The Ruling Class

Emperor – (EPR) The Emperor has the combined moves of King with an Abbey overlaid. The Emperor is designated as the replacement for the King on a larger chessboard as the piece that will be captured for checkmate. His index is 6 4.

Movement/Capture Summary

1 Orthogonal Moves (Castle-like) Forward 2 Squares Backward 2 Squares Sideward 2 Squares

Diagonal Moves (Bishop-like) Forward 1 Squares Backward 1 Squares L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Ruling Class

Queen – (QUN) The Queen has the combined moves of a Castle and a Bishop. This is the most powerful piece in traditional chess and is also a very powerful piece in Expanded Chess. The moves of the Queen remain exactly the same as in traditional chess. Her index is 6 3.

Movement/Capture Summary

3 Orthogonal Moves (Castle-like) Forward Unlimited Backward Unlimited Sideward Unlimited

Diagonal Moves (Bishop-like) Forward Unlimited Backward Unlimited L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Ruling Class

Duke – (DUK) The Duke has the combined moves of an Armory and a Priest. His index is 6 2.

Movement/Capture Summary

4 Orthogonal Moves (Castle-like) Forward 3 Squares Backward 3 Squares Sideward 3 Squares

Diagonal Moves (Bishop-like) Forward 3 Squares Backward 3 Squares L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Ruling Class

King – (KNG) The King is captured in traditional chess for checkmate. The moves of the King remain exactly the same as in traditional chess. His index is 6 1.

Movement/Capture Summary

5 Orthogonal Moves (Castle-like) Forward 1 Square Backward 1 Square Sideward 1 Square

Diagonal Moves (Bishop-like) Forward 1 Square Backward 1 Square L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Architectural Class Fortress – (FTS) The Fortress has the combined moves of a Castle and a Guard. Its index is 5 6.

Movement/Capture Summary

6 Orthogonal Moves (Castle-like) Forward Unlimited Backward Unlimited Sideward Unlimited

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward 3,2 L Backward 3,2 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Architectural Class

Citadel – (CDL) The Citadel has the combined moves of an Armory and an Archer. Its index is 5 5.

Movement/Capture Summary

7 Orthogonal Moves (Castle-like) Forward 3 Squares Backward 3 Squares Sideward 3 Squares

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward 3,1 L Backward 3,1 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Architectural Class Cathedral – (CTL) The Cathedral has the combined moves of a Knight and an Abbey. Its index is 5 4.

Movement/Capture Summary

8 Orthogonal Moves (Castle-like) Forward 2 Squares Backward 2 Squares Sideward 2 Squares

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward 2,1 L Backward 2,1 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Architectural Class Castle – (CSL) The Castle has exactly the same moves as a Rook in traditional chess except that it does not castle with the Emperor. Its index is 5 3.

Movement/Capture Summary

9 Orthogonal Moves (Castle-like) Forward Unlimited Backward Unlimited Sideward Unlimited

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Architectural Class Armory – (ARM) The Armory moves forward, backward and sideward like a Castle but is only able to move up to 3 squares in any single move. Its index is 5 2.

Movement/Capture Summary

a Orthogonal Moves (Castle-like) Forward 3 Squares Backward 3 Squares Sideward 3 Squares

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Architectural Class

Abbey – (ABY) The Abbey moves forward, backward or sideward like a Castle but is only able to move up to 2 squares in any single move. Its index is 5 1.

Movement/Capture Summary

b Orthogonal Moves (Castle-like) Forward 2 Squares Backward 2 Squares Sideward 2 Squares

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Religious Class

Pope – (POP) The Pope has the combined moves of a Bishop and a Guard. His index is 4 6.

Movement/Capture Summary

c Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward Unlimited Backward Unlimited L-Type Moves (Knight-like) Forward 3,2 L Backward 3,2 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Religious Class

Archbishop – (ARB) The Archbishop has the combined moves of a Priest and an Archer. His index is 4 5.

Movement/Capture Summary

d Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward 3 Squares Backward 3 Squares L-Type Moves (Knight-like) Forward 3,1 L Backward 3,1 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Religious Class

Cardinal – (CRD) The Cardinal has the combined moves of a Nun and a Knight. His index is 4 4.

Movement/Capture Summary

e Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward 2 Squares Backward 2 Squares L-Type Moves (Knight-like) Forward 2,1 L Backward 2,1 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Religious Class

Bishop – (BSP) The Bishop has exactly the same moves as in traditional chess. His index is 4 3.

Movement/Capture Summary

f Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward Unlimited Backward Unlimited L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Religious Class

Priest – (PRS) The Priest moves forward and backward diagonally like a Bishop but is only able to move up to 3 squares in any single move. His index is 4 2.

Movement/Capture Summary

g Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward 3 Squares Backward 3 Squares L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Religious Class

Nun – (NUN) The Nun moves forward and backward diagonally like a Bishop but is able to move only up to 2 squares in any single move. Her index is 4 1.

Movement/Capture Summary

h Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward 2 Squares Backward 2 Squares L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Military Class

Marshall – (MSL) The Marshall has the combined moves of a Guard and an Archer. His

index is 3 6.

Movement/Capture Summary

i Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward 3,2 L & 3,1 L Backward 3,2 L & 3,1 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE The Military Class

Captain – (CPN) The Captain has the combined moves of a Guard and a Knight. His index is 3 5.

Movement/Capture Summary

j Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward 3,2 L & 2,1 L Backward 3,2 L & 2,1 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Military Class

Sergeant – (SGT) The Sergeant the combined moves of an Archer and a Knight. His index is 3 4.

Movement/Capture Summary

k Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward 3,1 L & 2,1 L Backward 3,1 L & 2,1 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Military Class

Guard – (GRD) The Guard has moves similar to a Knight with 3,2 L-type moves. His index is 3 3.

Movement/Capture Summary

l Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward 3,2 L Backward 3,2 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Military Class

Archer – (ARC) The Archer has moves similar to a Knight with 3,1 L-type moves. His index is 3 2.

Movement/Capture Summary

m Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward 3,1 L Backward 3,1 L

MOVEMENT/CAPTURE DIAGRAM

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DESCRIPTION OF PIECE

The Military Class

Knight – (KNT) The Knight has exactly the same moves as in traditional chess. His index is 3 1.

Movement/Capture Summary

n Orthogonal Moves (Castle-like) Forward None Backward None Sideward None

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward 2,1 L Backward 2,1 L

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 n

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 26 of 37

DESCRIPTION OF PIECE

The Weaponry Class

Catapult – (CTP) The Catapult has limited L-type moves in the backward direction. Its index is 2 6.

Movement/Capture Summary

o Orthogonal Moves (Castle-like) Forward Unlimited Backward None Sideward None

Diagonal Moves (Bishop-like) Forward Unlimited Backward None L-Type Moves (Knight-like) Forward None Backward 3,2 L

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 o

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 27 of 37

DESCRIPTION OF PIECE The Weaponry Class

Ram – (RAM) The Ram has limited L-type moves in the backward direction. Its index is 2 5.

Movement/Capture Summary

p Orthogonal Moves (Castle-like) Forward Unlimited Backward None Sideward None

Diagonal Moves (Bishop-like) Forward 3 Squares Backward None L-Type Moves (Knight-like) Forward None Backward 3,1 L

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 p

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 28 of 37

DESCRIPTION OF PIECE The Weaponry Class

Mace – (MAC) The Mace has limited L-type moves in the backward direction .Its index is 2 4.

Movement/Capture Summary

q Orthogonal Moves (Castle-like) Forward 3 Squares Backward None Sideward None

Diagonal Moves (Bishop-like) Forward 2 Squares Backward None L-Type Moves (Knight-like) Forward None Backward 2,1 L

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 q

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 29 of 37

DESCRIPTION OF PIECE The Weaponry Class

Lance - (LNC) The Lance has limited L-type moves in the backward direction Its index is 2 3.

Movement/Capture Summary

r Orthogonal Moves (Castle-like) Forward Unlimited Backward None Sideward None

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward None Backward 3,2 L

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 r

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 30 of 37

DESCRIPTION OF PIECE The Weaponry Class

Hatchet – (HCT) The Hatchet has limited L-type moves in the backward direction. Its index is 2 2.

Movement/Capture Summary

s Orthogonal Moves (Castle-like) Forward 3 Squares Backward None Sideward None

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward None Backward 3,1 L

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 s

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 31 of 37

DESCRIPTION OF PIECE The Weaponry Class

Sword – (SWD) The Mace has limited L-type moves in the backward direction. Its index is 2 1.

Movement/Capture Summary

t Orthogonal Moves (Castle-like) Forward 2 Squares Backward None Sideward None

Diagonal Moves (Bishop-like) Forward None Backward None L-Type Moves (Knight-like) Forward None Backward 2,1 L

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 t

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 32 of 37

DESCRIPTION OF PIECE

The Special Character Class

Angel – (AGL) The Angel has very limited movement in the backward direction. Its index is 1 6.

Movement/Capture Summary

u Orthogonal Moves (Castle-like) Forward Unlimited Backward 3 Squares Sideward Unlimited

Diagonal Moves (Bishop-like) Forward Unlimited Backward None L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 u

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 33 of 37

DESCRIPTION OF PIECE The Special Character Class

Saint – (SNT) The Saint has very limited movement in the backward direction. Its index is 1 5.

Movement/Capture Summary

v Orthogonal Moves (Castle-like) Forward 3 Squares Backward 2 squares Sideward 3 Squares

Diagonal Moves (Bishop-like) Forward Unlimited Backward None L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 v

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 34 of 37

DESCRIPTION OF PIECE The Special Character Class

Wizard – (WZD) The Wizard has very limited movement in the backward direction. His index is 1 4.

Movement/Capture Summary

w Orthogonal Moves (Castle-like) Forward 3 Squares Backward 1 Square Sideward 3 Squares

Diagonal Moves (Bishop-like) Forward 3 Squares Backward None L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 w

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix A

Page 35 of 37

DESCRIPTION OF PIECE

The Special Character Class

Squire – (SQR) The Squire replaces the Pawn on larger game boards, only moves in the forward direction, but cannot execute En Passant. His index is 1 3.

Movement/Capture Summary

x Orthogonal Moves (Castle-like) Forward 3 Squares Backward None Sideward None

Diagonal Moves (Bishop-like) Forward 3 Squares Backward None L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 x

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

Forward Movement Only Promotion Piece

EXPANDED CHESS Appendix A

Page 36 of 37

DESCRIPTION OF PIECE

The Special Character Class

Knave (KNV) The Knave replaces the Pawn on larger game boards, only moves in the forward direction, but cannot execute En Passant. His index is 1 2.

Movement/Capture Summary

y Orthogonal Moves (Castle-like) Forward 2 Squares Backward None Sideward None

Diagonal Moves (Bishop-like) Forward 2 Squares Backward None L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 y

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

Forward Movement Only Promotion Piece

EXPANDED CHESS Appendix A

Page 37 of 37

DESCRIPTION OF PIECE The Special Character Class

Pawn – (PWN) The Pawn has exactly the same moves as in traditional chess and is able to move 1 square straight forward (up to 2 squares on opening move) and capture one square diagonal forward. The Pawn can also execute En Passant. His index is 1 6.

Movement/Capture Summary

z Orthogonal Moves (Castle-like) Forward 1 Square, 2 Squares*

first move only, optional Backward & Sideward None

Diagonal Moves (Bishop-like) Forward 1 Square (capture only) Backward None L-Type Moves (Knight-like) Forward None Backward None

MOVEMENT/CAPTURE DIAGRAM

16

15

14

13

12

11

10

09

08 z

07

06

05

04

03

02

01

a b c d e f g h i j k l m n o p

Forward Movement Only Promotion Piece

EXPANDED CHESS Appendix B Page 1 of 7

Expanded Chess Appendix B

DESCRIPTION OF

PIECES ON

CLASS BASIS

021345 6789ab cdefgh ijklmn opqrst uvwxyz

EXPANDED CHESS Appendix B Page 2 of 7

Description of Pieces on a Class Basis

The following description of the new types of pieces details these pieces on a class-by-class basis.

The Ruling Class

0 21 3 4 5

The Ruling Class originates from the obvious observation that the object of chess is to capture the King who is the leader of the Ruling Class. The Queen is the prototype piece for this class. This is the highest ranking class with extremely mobile and powerful pieces. The pieces include:

Piece Abbreviation Index

Empress (EPS) 6 6 Chancellor (CHL) 6 5 Emperor (EPR) 6 4 Queen (QUN) 6 3 Duke (DUK) 6 2 King (KNG) 6 1

EXPANDED CHESS Appendix B Page 3 of 7

The Architecture Class

6 7 8 9 a b

The Architecture Class originates from the observation that there is an architectural component to Chess. The Castle is the prototype piece for this class. This is the fifth ranking class with highly mobile and powerful pieces. The pieces include:

Piece Abbreviation Index

Fortress (FTS) 5 6 Citadel (CDL) 5 5 Cathedral (CTL) 5 4 Castle (CSL) 5 3 Armory (ARM) 5 2 Abbey (ABY) 5 1

EXPANDED CHESS Appendix B Page 4 of 7

The Religious Class

c d e f g h

The Religious Class originates from another observation that there is a religious component to Chess. The Bishop is the prototype piece for this class. This is the fourth ranking class with moderately mobile and powerful pieces. The pieces include:

Piece Abbreviation Index

Pope (POP) 4 6 Archbishop (ARB) 4 5 Cardinal (CRD) 4 4 Bishop (BSP) 4 3 Priest (PRS) 4 2 Nun (NUN) 4 1

EXPANDED CHESS Appendix B Page 5 of 7

The Military Class

i j k l m n

The Military Class originates from the also obvious observation that there is a military component to Chess. The Knight is the prototype piece for this class. This is the third ranking class with moderately mobile and powerful pieces. The pieces include:

Piece Abbreviation Index

Marshall (MSL) 3 6 Captain (CPN) 3 5 Sergeant (SGT) 3 4 Guard (GRD) 3 3 Archer (ARC) 3 2 Knight (KNT) 3 1

EXPANDED CHESS Appendix B Page 6 of 7

The Weaponry Class

o p q r s t

The Weaponry Class originates again from the observation that there is a military component to Chess and this class adds depth and choice to Expanded Chess. The Weaponry Class is newly added. This is the second ranking class with pieces that are designed for front line and near front line placement. The pieces include:

Piece Abbreviation Index

Catapult (CTP) 2 6 Ram (RAM) 2 5 Mace (MAC) 2 4 Lance (LNC) 2 3 Hatchet (HCT) 2 2 Sword (SWD) 2 1

EXPANDED CHESS Appendix B Page 7 of 7

The Special Characters Class

u v w x y z

The Special Characters Class originates from the final observation that there is a special piece in Chess. The Pawn is the prototype for the Special Characters Class. The special characteristics of the Pawn include:

Front line placement

Straight forward movement only

Diagonal forward capture only

Promotion ability

Lowest rating and considered sacrificial

The Angel, Saint and Wizard are included as special characters to add variety and have movements that are similar in structure to a Pawn but not as restrictive. The Squire, Knave and Pawn are designed for front line placement, cannot move backwards and can be promoted. This is the lowest ranking class. The pieces include:

Piece Abbreviation Index

Angel (AGL) 1 6 Saint (SNT) 1 5 Wizard (WZD) 1 4 Squire (SQR) 1 3 Knave (KNV) 1 2 Pawn (PWN) 1 1 Appendix A presents a detailed description of each piece including the image of each piece and its movements/capture diagram.

EXPANDED CHESS Appendix C Page 1 of 29

Expanded Chess Appendix C

GAME DESCRIPTION &

GAME ARRANGEMENT for

NEW GAMES

EXPANDED CHESS Appendix C Page 2 of 29

Table C-1 New Expanded Chess Games

Game Description & Game Arrangement

Game No. Size

(RxC)

No. of Pieces per

side L/I/H/C

No. of Pieces per side

Total

No. of Players per side

Checkmate Piece (King

Emperor)

Rows occupied @ start of game per

side

Table Name

0909-001 9 x 9 9/2/6/1 18 2 King 2 C-2 Perfect Chess

1010-001 10 x 10 10/2/7/1 20 2 King 2 C-3 Breakout Chess

1111-001 11 x 11 11/6/15/1 33 2 Emperor 3 C-4 Next Level

Chess

1212-001 12 x 12 12/12/11/1 36 2 Emperor 3 C-5 Arrival Chess

1313-001 13 x 13 13/16/9/1 39 2 Emperor 3 C-6 Lucky 13 Chess

1414-001 14 x 14 14/28/13/1 56 2 Emperor 4 C-7 Steinitz Chess

1515-001 15 x 15 15/31/13/1 60 2 Emperor 4 C-8 Laskers Chess

1616-001 16 x 16 16/32/15/1 64 2 Emperor 4 C-9 Tarrasch Chess

EXPANDED CHESS Appendix C Page 3 of 29

Expanded Chess Game Sheet 1A

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Traditional FIDE Chess

2 Game Identification Number 0808-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 8

2 Dimensions of the game board – columns deep 8

3 Number of rows occupied at the start of the game per side 2

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 5

2 Total number of types of High level pieces 3 39f

3 Total number of types of Intermediate level pieces 1 n

4 Total number of types of Low level pieces 1 z

5 Total number of types of pieces per side 6

6 Total number of Checkmate level pieces 1 5

7 Total number of High level pieces 5 3 99 ff

8 Total number of Intermediate level pieces 2 nn

9 Total number of Low level pieces 8 zzzzzzzz

10 Total number of pieces per side 16

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 1

3 Multiple moves with any piece Not Included

4 Multiple moves with single piece Not Included

5 Multiple pieces with single move each Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Included

2 En Passant Included

3 Castling Included

4 Slide Bishop Not Included

EXPANDED CHESS Appendix C Page 4 of 29

Expanded Chess Game Sheet 1B

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Traditional FIDE Chess

2 Game Identification Number 0808-001

1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote z

2 Pieces available for promotion

2a Direct Promotion 39f n 2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4 Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game e01 e08

EXPANDED CHESS Appendix C Page 5 of 29

Expanded Chess Game Sheet 1C Game Name Traditional FIDE Chess

Game Identification Number 0808-001

16x16 Game Board

16

15

14

13

12

11

10

09

08 9 n f 3 5 f n 9

07 z z z z z z z z

06

05

04

03

02 z z z z z z z z

01 9 n f 3 5 f n 9

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix C Page 6 of 29

Expanded Chess Game Sheet 2A

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Perfect Chess

2 Game Identification Number 0909-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 9

2 Dimensions of the game board – columns deep 9

3 Number of rows occupied at the start of the game per side 2

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 5 2 Total number of types of High level pieces 3 39f 3 Total number of types of Intermediate level pieces 1 n 4 Total number of types of Low level pieces 1 z 5 Total number of types of pieces per side 6

6 Total number of Checkmate level pieces 1 5 7 Total number of High level pieces 6 33 99 ff 8 Total number of Intermediate level pieces 2 nn 9 Total number of Low level pieces 9 zzzzzzzzz 10 Total number of pieces per side 18

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 1

3 Multiple Moves with Different Piece Each Move Not Included

4 Multiple Moves with Same Piece Each Move Not Included

5 Multiple Moves with Any Piece Each Move Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Included

2 En Passant Not Included

3 Castling Included

4 Slide Bishop Included

EXPANDED CHESS Appendix C Page 7 of 29

Expanded Chess Game Sheet 2B

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Perfect Chess

2 Game Identification Number 0909-001

1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote z

2 Pieces available for promotion

2a Direct Promotion 39f n 2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4 Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game e01 e09

EXPANDED CHESS Appendix C Page 8 of 29

Expanded Chess Game Sheet 2C Game Name Perfect Chess

Game Identification Number 0909-001

16x16 Game Board

16

15

14

13

12

11

10

09 9 n f 3 5 3H3

f n 9

08 z z z z z z z z z

07

06

05

04

03

02 z z z z z z z z z

01 9 n f 3 5 3H3

f n 9

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix C Page 9 of 29

Expanded Chess Game Sheet 3A

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Breakout Chess

2 Game Identification Number 1010-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 10

2 Dimensions of the game board – columns deep 10

3 Number of rows occupied at the start of the game per side 2

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 5

2 Total number of types of High level pieces 4 39of

3 Total number of types of Intermediate level pieces 1 n

4 Total number of types of Low level pieces 1 y

5 Total number of types of pieces per side 7

6 Total number of Checkmate level pieces 1 5

7 Total number of High level pieces 7 3 99 oo ff

8 Total number of Intermediate level pieces 2 nn

9 Total number of Low level pieces 10 yyyyyyyyyy

10 Total number of pieces per side 20

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 1

3 Multiple Moves with Different Piece Each Move Not Included

4 Multiple Moves with Same Piece Each Move Not Included

5 Multiple Moves with Any Piece Each Move Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Not Applicable

2 En Passant Not Applicable

3 Castling Included

4 Slide Bishop Not Included

EXPANDED CHESS Appendix C Page 10 of 29

Expanded Chess Game Sheet 3B

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Breakout Chess

2 Game Identification Number 1010-001

1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote y

2 Pieces available for promotion

2a Direct Promotion 39of n 2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4 Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game f01 f10

EXPANDED CHESS Appendix C Page 11 of 29

Expanded Chess Game Sheet 3C Game Name Breakout Chess

Game Identification Number 1010-001

16x16 Game Board

16

15

14

13

12

11

10 6 n 2o

f 15

3 f 2 n 6

09 y yz

y y yz

yz

yz

yz

yz

y

08

07

06

05

04

03

02 y yz

y y yz

yz

yz

yz

yz

y

01 6 n 2o

f 15

3 f 2 n 6

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix C Page 12 of 29

Expanded Chess Game Sheet 4A

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Next Level Chess

2 Game Identification Number 1111-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 11

2 Dimensions of the game board – columns deep 11

3 Number of rows occupied at the start of the game per side 3

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 1

2 Total number of types of High level pieces 5 369fu

3 Total number of types of Intermediate level pieces 5 nim7p

4 Total number of types of Low level pieces 1 y

5 Total number of types of pieces per side 13

6 Total number of Checkmate level pieces 1 1

7 Total number of High level pieces 9 33 66 9 uu ff

8 Total number of Intermediate level pieces 12 nn ii mm 77 pppp

9 Total number of Low level pieces 11 yyyyyyyyyyy

10 Total number of pieces per side 33

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 1

3 Multiple Moves with Different Piece Each Move Not Included

4 Multiple Moves with Same Piece Each Move Not Included

5 Multiple Moves with Any Piece Each Move Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Not Applicable

2 En Passant Not Applicable

3 Castling Not Included

4 Slide Bishop Not Included

EXPANDED CHESS Appendix C Page 13 of 29

Expanded Chess Game Sheet 4B

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Next Level Chess

2 Game Identification Number 1111-001

1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote y

2 Pieces available for promotion

2a Direct Promotion 369fu nim7p 2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4 Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game f01 f11

EXPANDED CHESS Appendix C Page 14 of 29

Expanded Chess Game Sheet 4C Game Name Next Level Chess

Game Identification Number 1111-001

16x16 Game Board

16

15

14

13

12

11 63H1

i6

ni

f23H13

3410

1000y

C1

30H4

f23H5

ni

i6

63H6

10 7I1

pm

26534pm

m u 93

u m 2 p 76

09 y yz

y y yz

yz

yz

yz

yz

y y

08

07

06

05

04

03 y y y y yz

yz

yz

yz

yz

y y

02 7I1

pm

2pm

m u 93

u m 2p

p 76

01 63H1

i6

ni

f23H13

3410

1000y

C1

30H4

f23H5

ni

i6

63H6

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix C Page 15 of 29

Expanded Chess Game Sheet 5A

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Arrival Chess

2 Game Identification Number 1212-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 12

2 Dimensions of the game board – columns deep 12

3 Number of rows occupied at the start of the game per side 3

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 1

2 Total number of types of High level pieces 6 0369cf

3 Total number of types of Intermediate level pieces 6 j2 a q k v

4 Total number of types of Low level pieces 1 y

5 Total number of types of pieces per side 14

6 Total number of Checkmate level pieces 1 1

7 Total number of High level pieces 11 0336699ccff

8 Total number of Intermediate level pieces 12 jj 22 aa qq kk vv

9 Total number of Low level pieces 12 yyyyyyyyyyyy

10

Total number of pieces per side 36

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 1

3 Multiple Moves with Different Piece Each Move Not Included

4 Multiple Moves with Same Piece Each Move Not Included

5 Multiple Moves with Any Piece Each Move Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Not Applicable

2 En Passant Not Applicable

3 Castling Not Included

4 Slide Bishop Not Included

EXPANDED CHESS Appendix C Page 16 of 29

Expanded Chess Game Sheet 5B

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Arrival Chess

2 Game Identification Number 1212-001

1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote y

2 Pieces available for promotion

2a Direct Promotion 0369cf j2aqkv 2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4 Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game f01 f12

EXPANDED CHESS Appendix C Page 17 of 29

Expanded Chess Game Sheet 5C Game Name Arrival Chess

Game Identification Number 1212-001

16x16 Game Board

16

15

14

13

12 63H1

j 2 c 3 1 0 3 c 2 j 63H1

11 a q k v 9 f f 9 v k q a

10 y yz

y y yz

yz

yz

yz

yz

y y y

09

08

07

06

05

04

03 y yz

y y yz

yz

y yz

yz

y y y

02 a q k v 9 f f 9 v k q a

01 63H1

j 2 c 3 1 0 3 c 2 j 63H1

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix C Page 18 of 29

Expanded Chess Game Sheet 6A

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Lucky 13 Chess

2 Game Identification Number 1313-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 13

2 Dimensions of the game board – columns deep 13

3 Number of rows occupied at the start of the game per side 3

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 1

2 Total number of types of High level pieces 5 03c6u

3 Total number of types of Intermediate level pieces 7 4i8rmgl

4 Total number of types of Low level pieces 2 yx

5 Total number of types of pieces per side 15

6 Total number of Checkmate level pieces 1 1

7 Total number of High level pieces 9 0033cc66u

8 Total number of Intermediate level pieces

16

44ii88rrmmggllll

9 Total number of Low level pieces 13 yyyyyyyyyyxxx

10 Total number of pieces per side 39

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 1

3 Multiple Moves with Different Piece Each Move Not Included

4 Multiple Moves with Same Piece Each Move Not Included

5 Multiple Moves with Any Piece Each Move Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Not Applicable

2 En Passant Not Applicable

3 Castling Not Included

4 Slide Bishop Not Included

EXPANDED CHESS Appendix C Page 19 of 29

Expanded Chess Game Sheet 6B

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Lucky 13 Chess

2 Game Identification Number 1313-001

1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote

yx 2 Pieces available for promotion

2a Direct Promotion 03c6u 4i8rmgl 2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4

Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game g01 g13

EXPANDED CHESS Appendix C Page 20 of 29

Expanded Chess Game Sheet 6C Game Name Lucky 13 Chess

Game Identification Number 1313-001

16x16 Game Board

16

15

14

13 6 i 4 c 3 0 1 0 3 c 4 i 6

12 8 prm

r

m g 2l

l u9

l 2l

g m putusqsrm

8

11 y yz

y y yz

x x x yz

y y y y

10

09

08

07

06

05

04

03 y yz

y y yz

x x x yz

y y y y

02 8 pr

m g 2 l u9

l 2 g m pr

8

01 6 i 4 c 3 0 1 0 3 c 4 i 6

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix C Page 21 of 29

Expanded Chess Game Sheet 7A

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Steinitz Chess

2 Game Identification Number 1414-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 14

2 Dimensions of the game board – columns deep 14

3 Number of rows occupied at the start of the game per side 4

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 1

2 Total number of types of High level pieces 7 03f96uo

3 Total number of types of Intermediate level pieces 9 ingad48ml

4 Total number of types of Low level pieces 2 yx

5 Total number of types of pieces per side 19

6 Total number of Checkmate level pieces 1 1

7 Total number of High level pieces 5 03f96uo

8 Total number of Intermediate level pieces 2 ingad48ml

9 Total number of Low level pieces 8 yx

10 Total number of pieces per side 56

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 2

3 Multiple Moves with Different Piece Each Move Not Included

4 Multiple Moves with Same Piece Each Move Not Included

5 Multiple Moves with Any Piece Each Move Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Not Applicable

2 En Passant Not Applicable

3 Castling Not Included

4 Slide Bishop Not Included

EXPANDED CHESS Appendix C Page 22 of 29

Expanded Chess Game Sheet 7B

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Steinitz Chess

2 Game Identification Number 1414-001

1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote yx

2 Pieces available for promotion 2a Direct Promotion 03f96uo ngad48ml 2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4 Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game g01 g14

EXPANDED CHESS Appendix C Page 23 of 29

Expanded Chess Game Sheet 7C Game Name Steinitz Chess

Game Identification Number 1414-001

16x16 Game Board

16

15

14 6 n i 98678abcdef

c9

3 1 0 3 cf

9 i n 6

13 g a a d 2 f u u f 2 d a a g

12 8 m o l m l o o l m l o m 8

11 y yz

y x x x x x x x x y yz

y

10

09

08

07

06

05

04 y yz

y x x x x x x x x y yz

y

03 8 m o l m l o o l m l o m 8

02 g a a d 2 f u u f 2 d a a g

01 6 n i 9c

c

9

3 1 0 3 c 9 i n 6

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix C Page 24 of 29

Expanded Chess Game Sheet 8A

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Laskers Chess

2 Game Identification Number 1515-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 15

2 Dimensions of the game board – columns deep 15

3 Number of rows occupied at the start of the game per side 4

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 1

2 Total number of types of High level pieces 8 03f6o9uc

3 Total number of types of Intermediate level pieces 12 10 j47hw2nepk

4 Total number of types of Low level pieces 2 yx

5 Total number of types of pieces per side 21

6 Total number of Checkmate level pieces 1 1

7 Total number of High level pieces 18 0033ff66oooo99uuc

8 Total number of Intermediate level pieces 27 jj444477hhwwww2nn eepppp kkkk

9 Total number of Low level pieces 15 yyyyyyyyxxxxxxx

10 Total number of pieces per side 60

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 2

3 Multiple Moves with Different Piece Each Move Not Included

4 Multiple Moves with Same Piece Each Move Not Included

5 Multiple Moves with Any Piece Each Move Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Not Applicable

2 En Passant Not Applicable

3 Castling Not Included

4 Slide Bishop Not Included

EXPANDED CHESS Appendix C Page 25 of 29

Expanded Chess Game Sheet 8B

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Laskers Chess

2 Game Identification Number 1515-001 1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote yx

2 Pieces available for promotion

2a Direct Promotion 03f6o9uc 2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4 Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game h01 h15

EXPANDED CHESS Appendix C Page 26 of 29

Table C-8B Expanded Chess Sheet 8C Game Name Laskers Chess

Game Identification Number 1515-001

16x16 Game Board

16

15 6 76

4 j

f 3 0 1 0 3 f j 4 76

6

14 h w w o o 4 9 2 9 4 o o w w h

13 n ef

p p k k u c u k k p p e n

12 y x y x y x y x y x y x y x y

11

10

09

08

07

06

05

04 y x y x y x y x y x y x y x y

03 n ef

p p k k u c u k k p p e n

02 h w w o o 4 9 2 9 4 o o w w h

01 6 76

4 j

f 3 0 1 0 3 f j 4 76

6

a b c d e f g h i j k l m n o p

EXPANDED CHESS Appendix C Page 27 of 29

Expanded Chess Game Sheet 9A

0213456789abcdefghijklmnopqrstuvwxyz General Items

1 Game Name Tarrasch Chess

2 Game Identification Number 1616-001

1.1 Game Board Arrangement

1 Dimensions of the game board – rows wide 16

2 Dimensions of the game board – columns deep 16

3 Number of rows occupied at the start of the game per side 4

4 Number of colors on the game board 2

1.2 Game Material Configuration

1 Total number of types of Checkmate level pieces 1 1

2 Total number of types of High level pieces 8 03u69fco

3 Total number of types of Intermediate level pieces 11 2dhbg4smvji

4 Total number of types of Low level pieces 2 yx

5 Total number of types of pieces per side 22

6 Total number of Checkmate level pieces 1 1

7 Total number of High level pieces 180033uu6699ffffccoo

8 Total number of Intermediate level pieces 9222ddddhhbbgg44ssmmmm vvvvjjii

9 Total number of Low level pieces 16 yyyyxxxxxxxxxxxx

10

Total number of pieces per side 64

1.3 Players

1 Number of players per side 1

2 Number of moves in turn for each side 2

3 Multiple Moves with Different Piece Each Move Included

4 Multiple Moves with Same Piece Each Move Not Included

5 Multiple Moves with Any Piece Each Move Not Included

6 Player’s access to game board Unlimited

7 Player’s access to game board – limited by spatial boundaries Unlimited

8 Player’s access to game board – limited by square color Allowed to play: Black White

9 Player’s access to game material Unlimited

10 Player’s access to game material – limited to pieces shown N/A

1.4 Special Rules

1 Two Move – Pawn Opening Not Applicable

2 En Passant Not Applicable

3 Castling Not Included

4 Slide Bishop Not Included

EXPANDED CHESS Appendix C Page 28 of 29

Expanded Chess Game Sheet 9B

0213456789abcdefghijklmnopqrstuvwxyz General Items

1.5 Capture

1 Rewards of capture - Return of a player’s previously captured piece of equal or lesser strength to the game board on capture square.

Not Included

2 Rewards of capture - Move capturing piece to a new location and leave captured piece on board in present location.

Not Included

3 Penalties of capture - Remove captured piece from the game board for the duration of the game.

Included

4 Penalties of capture - Move captured piece to a new location chosen by capturing player.

Not Included

1.6 Promotion

1 Pieces each player can promote yx

2 Pieces available for promotion

2a Direct Promotion 03u69fco2dhbg4smvji

2b Promotion Reward Not Included

1.7 Check

1

Rewards of check - Having an already captured low or intermediate level piece returned to the game in lieu of forcing the checked piece to move or blocking the check. The returned piece will be placed on the location where the check would have occurred.

Not Included

2

Rewards of check - Place checking piece to a new location after declaring check and essentially using check to gain a free move and leave checked piece on board in present location. The new location may or may not be a checking location.

Not Included

3

Penalties of check - Removing another piece in lieu of forcing the checked piece to move or blocking the check. The checking piece would stay at its pre-check location after declaring the intention to check and the location of the checking piece, and then remove the opponent’s piece from the board. Removal of low or intermediate level pieces only will be permitted.

Not Included

4

Penalties of check - Preventing a potential check-blocking piece from moving to a blocking location and forcing Checkmate piece to move instead.

Not Included

1.8 Checkmate

1 Checkmate piece location at start of game i01 i08

EXPANDED CHESS Appendix C Page 29 of 29

Expanded Chess Game Sheet 9C Game Name Tarrasch Chess

Game Identification Number 1616-001

16x16 Game Board

16 9 6 d d u 3 0 2 1 0 3 u d d 6 9 15 h b g 4 f f 2 c c 2 f f 4 g b h 14 s m m v v o j i i j o v v m m s 13 x x y x x y x x x x y x x y x x 12

11

10

09

08

07

06

05

04 x x y x x y x x x x y x x y x x 03 s m m v v o j i i j o v v m m s 02 h b g 4 f f 2 c c 2 f f 4 g b h 01 9 6 d d u 3 0 2 1 0 3 u d d 6 9

a b c d e f g h i j k l m n o p

Types/Components of Movements Summary Matrix

START OF GAME

Piece R C ABR 7.50 3 2 1 0 7.50 3 2 1 0 7.50 3 2 1 0 7.00 3 2 1 0 4,2 3,2 4,1 3,1 2,1 0 4,2 3,2 4,1 3,1 2,1 0 4,2 3,2 4,1 3,1 2,1 0 4,2 3,2 4,1 3,1 2,1 0

Empress 6 6 EPS x x x x x x x x

Chancellor 6 5 CHL x x x x x x x x

Emperor 6 4 EPR x x x x x x x x

Queen 6 3 QUN x x x x x x x x

Duke 6 2 DUK x x x x x x x x

King 6 1 KNG x x x x x x x x

Fortress 5 6 FTS x x x x x x x x

Citadel 5 5 CDL x x x x x x x x

Cathedral 5 4 CTL x x x x x x x x

Castle 5 3 CSL x x x x x x x x

Armory 5 2 ARM x x x x x x x x

Abbey 5 1 ABY x x x x x x x x

Pope 4 6 POP x x x x x x x x

Archbishop 4 5 ARB x x x x x x x x

Cardinal 4 4 CRD x x x x x x x x

Bishop 4 3 BSP x x x x x x x x

Priest 4 2 PRS x x x x x x x x

Nun 4 1 NUN x x x x x x x x

Marshall 3 6 MSL x x x x x x x x

Captain 3 5 CPN x x x x x x x x

Sergeant 3 4 SGT x x x x x x x x

Guard 3 3 GRD x x x x x x x x

Archer 3 2 ARC x x x x x x x x

Knight 3 1 KNT x x x x x x x x

Catapult 2 6 CTP f x x x x x x x

Ram 2 5 RAM f x x x x x x x

Mace 2 4 MAC f x x x x x x x

Lance 2 3 LNC f x x x x x x x

Hatchet 2 2 HCT f x x x x x x x

Sword 2 1 SWD f x x x x x x x

Angel 1 6 AGL f b x x x x x x x

Saint 1 5 SNT f b x x x x x x x

Wizard 1 4 WZD f b x x x x x x x

Squire 1 3 SQR f x x x x x x x

Knave 1 2 KNV f x x x x x x x

Pawn 1 1 PWN f x x x x x x x

L2-Type F/LR L2-Type B/LRL1-Type F/LR L1-Type B/LROrthogonal F/B Orthogonal L/R Diagonal F/LR Diagonal B/LR

EXPANDED CHESS Appendix D1

1 of 1

Types/Components of Movements Summary Matrix

CLEAR BOARD

Piece R C ABR 7.5 3 2 1 0 7.5 3 2 1 0 7.5 3 2 1 0 7.0 3 2 1 0 4,2 3,2 4,1 3,1 2,1 0 4,2 3,2 4,1 3,1 2,1 0 4,2 3,2 4,1 3,1 2,1 0 4,2 3,2 4,1 3,1 2,1 0

Empress 6 6 EPS x x x x x x x x

Chancellor 6 5 EPR x x x x x x x x

Emperor 6 4 CHL x x x x x x x x

Queen 6 3 QUN x x x x x x x x

Duke 6 2 DUK x x x x x x x x

King 6 1 KNG x x x x x x x x

Fortress 5 6 FTS x x x x x x x x

Citadel 5 5 CDL x x x x x x x x

Cathedral 5 4 CTL x x x x x x x x

Castle 5 3 CSL x x x x x x x x

Armory 5 2 ARM x x x x x x x x

Abbey 5 1 ABY x x x x x x x x

Pope 4 6 POP x x x x x x x x

Archbishop 4 5 ARB x x x x x x x x

Cardinal 4 4 CRD x x x x x x x x

Bishop 4 3 BSP x x x x x x x x

Priest 4 2 PRS x x x x x x x x

Nun 4 1 NUN x x x x x x x x

Marshall 3 6 MSL x x x x x x x x

Captain 3 5 CPN x x x x x x x x

Sergeant 3 4 SGT x x x x x x x x

Guard 3 3 GRD x x x x x x x x

Archer 3 2 ARC x x x x x x x x

Knight 3 1 KNT x x x x x x x x

Catapult 2 6 CTP f x x x x x x x

Ram 2 5 RAM f x x x x x x x

Mace 2 4 MAC f x x x x x x x

Lance 2 3 LNC f x x x x x x x

Hatchet 2 2 HCT f x x x x x x x

Sword 2 1 SWD f x x x x x x x

Angel 1 6 AGL f b x x x x x x x

Saint 1 5 SNT f b x x x x x x x

Wizard 1 4 WZD f b x x x x x x x

Squire 1 3 SQR f x x x x x x x

Knave 1 2 KNV f x x x x x x x

Pawn 1 1 PWN f x x x x x x x

L2-Type B/LRL1-Type F/LR L1-Type B/LROrthogonal F/B Orthogonal L/R Diagonal F/LR Diagonal B/LR L2-Type F/LR

EXPANDED CHESS Appendix D2

1 of 1

Types/Components of Movements Summary Table

START OF GAME

L2 Coverage

SG

L1 Coverage

SG

Total Diag

Coverage

Total Orth

Coverage L2 range L1 range

Orth Weight

Range

Diag

Max Range

Orth L/R Max

Range

Orth F/B Max

Range

Piece

Empress 0.000 8.000 29.000 30.000 0.000 7.211 21.213 14.500 15.000 15.000

Chancellor 0.000 8.000 12.000 12.000 0.000 6.633 8.485 6.000 6.000 6.000

Emperor 0.000 0.000 4.000 8.000 0.000 0.000 5.657 2.000 4.000 4.000

Queen 0.000 0.000 29.000 30.000 0.000 0.000 21.213 14.500 15.000 15.000

Duke 0.000 0.000 12.000 12.000 0.000 0.000 8.485 6.000 6.000 6.000

King 0.000 0.000 4.000 4.000 0.000 0.000 2.828 2.000 2.000 2.000

Fortress 0.000 8.000 0.000 30.000 0.000 7.211 21.213 0.000 15.000 15.000

Citadel 0.000 8.000 0.000 12.000 0.000 6.633 8.485 0.000 6.000 6.000

Cathedral 0.000 8.000 0.000 8.000 0.000 4.472 5.657 0.000 4.000 4.000

Castle 0.000 0.000 0.000 30.000 0.000 0.000 21.213 0.000 15.000 15.000

Armory 0.000 0.000 0.000 12.000 0.000 0.000 8.485 0.000 6.000 6.000

Abbey 0.000 0.000 0.000 8.000 0.000 0.000 5.657 0.000 4.000 4.000

Pope 0.000 8.000 29.000 0.000 0.000 7.211 0.000 14.500 0.000 0.000

Archbishop 0.000 8.000 12.000 0.000 0.000 6.633 0.000 6.000 0.000 0.000

Cardinal 0.000 8.000 6.000 0.000 0.000 4.472 0.000 3.000 0.000 0.000

Bishop 0.000 0.000 29.000 0.000 0.000 0.000 0.000 14.500 0.000 0.000

Priest 0.000 0.000 12.000 0.000 0.000 0.000 0.000 6.000 0.000 0.000

Nun 0.000 0.000 6.000 0.000 0.000 0.000 0.000 3.000 0.000 0.000

Marshall 8.000 8.000 0.000 0.000 6.633 7.211 0.000 0.000 0.000 0.000

Captain 8.000 8.000 0.000 0.000 4.472 7.211 0.000 0.000 0.000 0.000

Sergeant 8.000 8.000 0.000 0.000 4.472 6.633 0.000 0.000 0.000 0.000

Guard 0.000 8.000 0.000 0.000 0.000 7.211 0.000 0.000 0.000 0.000

Archer 0.000 8.000 0.000 0.000 0.000 6.633 0.000 0.000 0.000 0.000

Knight 0.000 8.000 0.000 0.000 0.000 4.472 0.000 0.000 0.000 0.000

Catapult 0.000 4.000 15.000 7.500 0.000 3.606 7.500 7.500 0.000 7.500

Ram 0.000 4.000 6.000 7.500 0.000 3.317 7.500 3.000 0.000 7.500

Mace 0.000 4.000 4.000 3.000 0.000 2.236 3.000 2.000 0.000 3.000

Lance 0.000 4.000 0.000 7.500 0.000 3.606 7.500 0.000 0.000 7.500

Hatchet 0.000 4.000 0.000 3.000 0.000 3.317 3.000 0.000 0.000 3.000

Sword 0.000 4.000 0.000 2.000 0.000 2.236 2.000 0.000 0.000 2.000

Angel 0.000 0.000 15.000 22.500 0.000 0.000 16.771 7.500 15.000 7.500

Saint 0.000 0.000 15.000 9.000 0.000 0.000 6.708 7.500 6.000 3.000

Wizard 0.000 0.000 6.000 9.000 0.000 0.000 6.708 3.000 6.000 3.000

Squire 0.000 0.000 6.000 3.000 0.000 0.000 3.000 3.000 0.000 3.000

Knave 0.000 0.000 4.000 2.000 0.000 0.000 2.000 2.000 0.000 2.000

Pawn 0.000 0.000 0.666 1.000 0.000 0.000 1.200 0.333 0.000 1.200

EXPANDED CHESS Appendix E

1 of 2

Types/Components of Movements Summary Table

CLEAR BOARD

L2 Coverage

CB

L1 Coverage

CB

Total Diag

Coverage

Total Orth

Coverage L2 range L1 range

Orth Weight

Range

Diag

Max Range

Orth L/R Max

Range

Orth F/B Max

Range

Piece

Empress 0.000 8.000 29.000 30.000 0.000 7.211 21.213 14.500 15.000 15.000

Chancellor 0.000 8.000 12.000 12.000 0.000 6.633 8.485 6.000 6.000 6.000

Emperor 0.000 0.000 4.000 8.000 0.000 0.000 5.657 2.000 4.000 4.000

Queen 0.000 0.000 29.000 30.000 0.000 0.000 21.213 14.500 15.000 15.000

Duke 0.000 0.000 12.000 12.000 0.000 0.000 8.485 6.000 6.000 6.000

King 0.000 0.000 4.000 4.000 0.000 0.000 2.828 2.000 2.000 2.000

Fortress 0.000 8.000 0.000 30.000 0.000 7.211 21.213 0.000 15.000 15.000

Citadel 0.000 8.000 0.000 12.000 0.000 6.633 8.485 0.000 6.000 6.000

Cathedral 0.000 8.000 0.000 8.000 0.000 4.472 5.657 0.000 4.000 4.000

Castle 0.000 0.000 0.000 30.000 0.000 0.000 21.213 0.000 15.000 15.000

Armory 0.000 0.000 0.000 12.000 0.000 0.000 8.485 0.000 6.000 6.000

Abbey 0.000 0.000 0.000 8.000 0.000 0.000 5.657 0.000 4.000 4.000

Pope 0.000 8.000 29.000 0.000 0.000 7.211 0.000 14.500 0.000 0.000

Archbishop 0.000 8.000 12.000 0.000 0.000 6.633 0.000 6.000 0.000 0.000

Cardinal 0.000 8.000 6.000 0.000 0.000 4.472 0.000 3.000 0.000 0.000

Bishop 0.000 0.000 29.000 0.000 0.000 0.000 0.000 14.500 0.000 0.000

Priest 0.000 0.000 12.000 0.000 0.000 0.000 0.000 6.000 0.000 0.000

Nun 0.000 0.000 6.000 0.000 0.000 0.000 0.000 3.000 0.000 0.000

Marshall 8.000 8.000 0.000 0.000 6.633 7.211 0.000 0.000 0.000 0.000

Captain 8.000 8.000 0.000 0.000 4.472 7.211 0.000 0.000 0.000 0.000

Sergeant 8.000 8.000 0.000 0.000 4.472 6.633 0.000 0.000 0.000 0.000

Guard 0.000 8.000 0.000 0.000 0.000 7.211 0.000 0.000 0.000 0.000

Archer 0.000 8.000 0.000 0.000 0.000 6.633 0.000 0.000 0.000 0.000

Knight 0.000 8.000 0.000 0.000 0.000 4.472 0.000 0.000 0.000 0.000

Catapult 0.000 4.000 15.000 7.500 0.000 3.606 7.500 7.500 0.000 7.500

Ram 0.000 4.000 6.000 7.500 0.000 3.317 7.500 3.000 0.000 7.500

Mace 0.000 4.000 4.000 3.000 0.000 2.236 3.000 2.000 0.000 3.000

Lance 0.000 4.000 0.000 7.500 0.000 3.606 7.500 0.000 0.000 7.500

Hatchet 0.000 4.000 0.000 3.000 0.000 3.317 3.000 0.000 0.000 3.000

Sword 0.000 4.000 0.000 2.000 0.000 2.236 2.000 0.000 0.000 2.000

Angel 0.000 0.000 15.000 22.500 0.000 0.000 16.771 7.500 15.000 7.500

Saint 0.000 0.000 15.000 9.000 0.000 0.000 6.708 7.500 6.000 3.000

Wizard 0.000 0.000 6.000 9.000 0.000 0.000 6.708 3.000 6.000 3.000

Squire 0.000 0.000 6.000 3.000 0.000 0.000 3.000 3.000 0.000 3.000

Knave 0.000 0.000 4.000 2.000 0.000 0.000 2.000 2.000 0.000 2.000

Pawn 0.000 0.000 2.000 1.000 0.000 0.000 1.000 1.000 0.000 1.000

EXPANDED CHESS Appendix E

2 of 2

Weighting/Rating Summary Matrix

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

D E F G H I J K L M N O P Q R S T U V W

Piece Empress Chancellor Emperor Queen Duke King Fortress Citadel Cathedral Castle Armory Abbey Pope Archbishop Cardinal Bishop Priest Nun

TOTAL 217.33 123.36 48.36 181.22 87.57 29.19 160.50 93.29 63.78 121.95 57.50 38.33 97.82 65.86 40.48 59.27 30.07 15.04

Empress Chancellor Emperor Queen Duke King Fortress Citadel Cathedral Castle Armory Abbey Pope ArchbishopCardinal Bishop Priest Nun

Adjustment Factor 22.968 13.037 5.110 19.151 9.255 3.085 16.962 9.859 6.740 12.888 6.077 4.051 10.338 6.960 4.278 6.263 3.178 1.589

Empress Chancellor Emperor Queen Duke King Fortress Citadel Cathedral Castle Armory Abbey Pope Archbishop Cardinal Bishop Priest Nun

100.00 56.76 22.25 83.39 40.29 13.43 73.85 42.92 29.35 56.11 26.46 17.64 45.01 30.30 18.63 27.27 13.84 6.92

Piece Table Index 6 6 6 5 6 4 6 3 6 2 6 1 5 6 5 5 5 4 5 3 5 2 5 1 4 6 4 5 4 4 4 3 4 2 4 1

CwLRSpBd Columns wide L/R Span of Board 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

RdFBSpBd Rows deep F/B Span of Board 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

TotalSqBd = CwLRSpBd * RdFBSpBd Total Squares on Board 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000

RwOcPrSG Rows Occupied/Player @ Start of Game 4.000

TotalOrCvPc Total Orthogonal Coverage of Piece 30.000 12.000 8.000 30.000 12.000 4.000 30.000 12.000 8.000 30.000 12.000 8.000 0.000 0.000 0.000 0.000 0.000 0.000

TotalDgCvPc Total Diagonal Coverage of Piece 29.000 12.000 4.000 29.000 12.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000 29.000 12.000 6.000 29.000 12.000 6.000

L1CvPcSG L1 Coverage of Piece @ Start of Game 8.000 8.000 0.000 0.000 0.000 0.000 8.000 8.000 8.000 0.000 0.000 0.000 8.000 8.000 8.000 0.000 0.000 0.000

L2CvPcSG L2 Coverage of Piece @ Start of Game 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

L1CvPcCB L1 Coverage of Piece w/ Clear Board 8.000 8.000 0.000 0.000 0.000 0.000 8.000 8.000 8.000 0.000 0.000 0.000 8.000 8.000 8.000 0.000 0.000 0.000

L2CvPcCB L2 Coverage of Piece w/ Clear Board 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

L1RgPc L1 Range of Piece 7.211 6.633 0.000 0.000 0.000 0.000 7.211 6.633 4.472 0.000 0.000 0.000 7.211 6.633 4.472 0.000 0.000 0.000

L2RgPc L2 Range of Piece 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxFBOrRgPc Maximum F/B Orthogonal Range of Piece 15.000 6.000 4.000 15.000 6.000 2.000 15.000 6.000 4.000 15.000 6.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxLROrRgPc Maximum L/R Orthogonal Range of Piece 15.000 6.000 4.000 15.000 6.000 2.000 15.000 6.000 4.000 15.000 6.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxDgRgPc Maximum Diagonal Range of Piece 14.500 6.000 2.000 14.500 6.000 2.000 0.000 0.000 0.000 0.000 0.000 0.000 14.500 6.000 3.000 14.500 6.000 3.000

WeightedOrRgPc Weighted Orthogonal Range of Piece 21.213 8.485 5.657 21.213 8.485 2.828 21.213 8.485 5.657 21.213 8.485 5.657 0.000 0.000 0.000 0.000 0.000 0.000

CvSG Coverage @ Start of Game

WgtOrCvSG Weighting Orthogonal Coverage 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

WgtDgCvSG Weighting Diagonal Coverage 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

WgtL1L2CvSG Weighting L1& L2 Coverage 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643

CvCB Coverage w/ Clear Board

WgtOrCvCB Weighting Orthogonal Coverage 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

WgtDgCvCB Weighting Diagonal Coverage 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

WgtL1L2CvCB Weighting L1& L2 Coverage 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476

RgSG Range @ Start of Game

WgtFBOrRgSG = WgtOrCvSG Weighting F/B Orthogonal Range 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128

WgtLROrRgSG Weighting L/R Orthogonal Range 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

WgtDgRgSG Weighting Diagonal Range 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428

WgtL1L2RgSG Weighting L1& L2 Range 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500

RgCB Range w/ Clear Board

WgtFBOrRgCB Weighting F/B Orthogonal Range 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

WgtLROrRgCB Weighting L/R Orthogonal Range 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

WgtDgRgCB = WgtDgCvCB Weighting Diagonal Range 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

WgtL1L2RgCB Weighting L1& L2 Range 1.009 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500

EXPANDED CHESS Appendix F

1 of 6

Weighting/Rating Summary Matrix

4

D E F G H I J K L M N O P Q R S T U V W

Piece Empress Chancellor Emperor Queen Duke King Fortress Citadel Cathedral Castle Armory Abbey Pope Archbishop Cardinal Bishop Priest Nun

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

CvSG Coverage @ Start of Game

RwOcPrSG = RwOrPrSG Rows Occupied/Player @ Start of Game 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000

PcPrSG = CwLRSpBd * RwOcPrSG Pieces/Player @ Start of Game 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000

RwClrSG = RdFBSpBd * RwOcSG Rows Clear @ Start of Game 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

RwOcSG = RwOcPrSG * 2 Rows Occupied @ Start of Game 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000

SqClrSG = RwOcPrSG * CwLRSpBd Squares Clear @ Start of Game 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000

SqOcSG = RwOcSG * CwLRSpBd Squares Occupied @ Start of Game 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000

SqOpSG = SClrSG - 1 Squares Open @ Start of Game 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000

CwSG = CwLRSpBd Cw @ Start of Game 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

RdSG = RwClrSG Rd @ Start of Game 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

RdCwLTEQGTFlag (If CwSg < RdSg, 0,1) Rd/Cw LT/EQ-GT Flag 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Rd EvenOddFlag (If RdSG = Ev, 0 / Od, 1) Rd Even/Odd Flag 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

DgFCv = Table A Diagonal Forward Coverage 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000

DgBCv = Table B Diagonal Backward Coverage 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

DgTotalCv = DgFCv + DgBCv Diagonal Total Coverage 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000

OrLRCv = CwSG - 1 Orthogonal L/R Coverage 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000

OrFBCv = RdSG -1 Orthogonal F/B Coverage 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000

OrTotalCv = OrLRCv + OrFBCv Orthogonal Total Coverage 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000

MaxOrFBRgPcSG = Table C Maximum Orthogonal F/B Range of Piece @ Start of Game 11.000 6.000 4.000 11.000 6.000 2.000 11.000 6.000 4.000 11.000 6.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxOrLRRgPcSG = MaxOrLRRg Maximum Orthogonal L/R Range of Piece @ Start of Game 15.000 6.000 4.000 15.000 6.000 2.000 15.000 6.000 4.000 15.000 6.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxDgRgPcSG = Table D Maximum Diagonal Range of Piece @ Start of Game 11.000 6.000 2.000 11.000 6.000 2.000 0.000 0.000 0.000 0.000 0.000 0.000 11.000 6.000 3.000 11.000 6.000 3.000

OrSqCvPcSG = MaxOrFBRgPcSG + MaxOrLRRgPcSG Orthogonal Squares Covered by Piece @ Start of Game 26.000 12.000 8.000 26.000 12.000 4.000 26.000 12.000 8.000 26.000 12.000 8.000 0.000 0.000 0.000 0.000 0.000 0.000

DgSqCvPcSG = MaxDgRgPcSG * 2 Diagonal Squares Covered by Piece @ Start of Game 22.000 12.000 4.000 22.000 12.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000 22.000 12.000 6.000 22.000 12.000 6.000

L1SqCvPcSG = L1CvPcSG L1 Squares Covered by Piece @ Start of Game 8.000 8.000 0.000 0.000 0.000 0.000 8.000 8.000 8.000 0.000 0.000 0.000 8.000 8.000 8.000 0.000 0.000 0.000

L2SqCvPcSG = L2CvPcSG L2 Squares Covered by Piece @ Start of Game 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

OrEffSG = OrSqCvPcSG/SqOpSG*WgtOrCvSG Orthogonal Effectiveness @ Start of Game 0.136 0.063 0.042 0.136 0.063 0.021 0.136 0.063 0.042 0.136 0.063 0.042 0.000 0.000 0.000 0.000 0.000 0.000

DgEffSG = DgSqCvPcSG/SqOpSG*WgtDgCvSg Diagonal Effectiveness @ Start of Game 0.115 0.063 0.021 0.115 0.063 0.021 0.000 0.000 0.000 0.000 0.000 0.000 0.115 0.063 0.031 0.115 0.063 0.031

L1Eff SG = L1SqCvPcSG / SqOPSG * WgtL1L2CvSG L1 Effectiveness @ Start of Game 0.069 0.069 0.000 0.000 0.000 0.000 0.069 0.069 0.069 0.000 0.000 0.000 0.069 0.069 0.069 0.000 0.000 0.000

L2EffSG = L2SqCvPcSG / SqOPSG * WgtL1L2CvSG L2 Effectiveness @ Start of Game 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

EffCvSG = OrEffSG + DgEffSg + L1EffSG + L2EffSG Effectiveness 0.320 0.194 0.063 0.251 0.126 0.042 0.205 0.132 0.111 0.136 0.063 0.042 0.184 0.132 0.100 0.115 0.063 0.031

FactorDWgt Factor D Weighting 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856

CvSG Coverage @ Start of Game

FactorD = EffCvSG * FactorDWgt Factor D 7.957 4.834 1.562 6.247 3.123 1.041 5.094 3.272 2.752 3.384 1.562 1.041 4.574 3.272 2.491 2.863 1.562 0.781

CvCB Coverage w/ Clear Board

SqClrCB = TotalSqBd Squares Clear w/ Clear Board 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000

SqOpenCB = SqClrCB - 1 Squares Open w/ Clear Board 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000

OrSqCvCB = Total OrCvPc Orthogonal Squares Covered w/ Clear Board 30.000 12.000 8.000 30.000 12.000 4.000 30.000 12.000 8.000 30.000 12.000 8.000 0.000 0.000 0.000 0.000 0.000 0.000

DgSqCvCB = TotalDgCvPc Diagonal Squares Covered w/ Clear Board 29.000 12.000 4.000 29.000 12.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000 29.000 12.000 6.000 29.000 12.000 6.000

L1SqCvCB = L1CvPcCB L1 Squares Covered w/ Clear Board 8.000 8.000 0.000 0.000 0.000 0.000 8.000 8.000 8.000 0.000 0.000 0.000 8.000 8.000 8.000 0.000 0.000 0.000

L2SqCvCB = L2CvPcCB L2 Squares Covered w Clear Board 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

OrEffCB = OrSqCvCB / SqOpenCB * WgtOrCvCB Orthogonal Efectiveness w/ Clear Board 0.118 0.047 0.031 0.118 0.047 0.016 0.118 0.047 0.031 0.118 0.047 0.031 0.000 0.000 0.000 0.000 0.000 0.000

DgEffCB = DGSqCvCB / SqOpenCB * WgtDgCvCB Diagonal Effectiveness w/ Clear Board 0.114 0.047 0.016 0.114 0.047 0.016 0.000 0.000 0.000 0.000 0.000 0.000 0.114 0.047 0.024 0.114 0.047 0.024

L1EffCB = L1SqCvCB / SqOpenCB * WgtL1L2CvCB L1 Effectiveness w/ Clear Board 0.078 0.078 0.000 0.000 0.000 0.000 0.078 0.078 0.078 0.000 0.000 0.000 0.078 0.078 0.078 0.000 0.000 0.000

L2EffCB = L2SqCvCB / SqOpenCB * WgtL1L2CvCB L2 Effectiveness w/ Clear Board 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

EffCvCB Effectiveness 0.309 0.172 0.047 0.231 0.094 0.031 0.195 0.125 0.109 0.118 0.047 0.031 0.191 0.125 0.101 0.114 0.047 0.024

FactorEWgt Factor E Weighting 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000

CvCB Coverage w/ Clear Board

FactorE = EffCvCB * FactorEWgt Factor E 9.272 5.154 1.412 6.941 2.824 0.941 5.860 3.742 3.272 3.529 1.412 0.941 5.742 3.742 3.036 3.412 1.412 0.706

EXPANDED CHESS Appendix F

2 of 6

Weighting/Rating Summary Matrix

4

D E F G H I J K L M N O P Q R S T U V W

Piece Empress Chancellor Emperor Queen Duke King Fortress Citadel Cathedral Castle Armory Abbey Pope Archbishop Cardinal Bishop Priest Nun

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

RgSG Range @ Start of Game

MaxOrFBRgPcSG = MaxDgRgPcSG Maximum Orthogonal F/B Range of Piece @ Start of Game 11.000 6.000 4.000 11.000 6.000 2.000 11.000 6.000 4.000 11.000 6.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxOrFBSpBdSG = RwClrSG Maximum Orthogonal F/B Span of Board @ Start of Game 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

MaxOrLRRgPcSG = MaxLROrRgPc Maximum Orthogonal L/R Range of Piece @ Start of Game 15.000 6.000 4.000 15.000 6.000 2.000 15.000 6.000 4.000 15.000 6.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxOrLRSpBdSG = CwLRSpBd Maximum Orthogonal L/R Span of Board @ Start of Game 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

MaxDgRgPcSG = MaxDgRgPcSG Maximum Diagonal Range of Piece @ Start of Game 11.000 6.000 2.000 11.000 6.000 2.000 0.000 0.000 0.000 0.000 0.000 0.000 11.000 6.000 3.000 11.000 6.000 3.000

MaxDgSpBdSG = RwClrSG Maximum Diagonal Span of Board @ Start of Game 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

MaxL1RgPcSG = Table E Maximum L1 Range of Piece @ Start of Game 7.211 6.633 0.000 0.000 0.000 0.000 7.211 6.633 4.472 0.000 0.000 0.000 7.211 6.633 4.472 0.000 0.000 0.000

MaxL2RgPcSG = Table F Maximum L2 Range of Piece @ Start of Game 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxLSpBdSG = Sqrt(MaxOrFBSpBdSg^2 + MaxOrLRSpBdSg^2) Maximum L Span of Board @ Start of Game 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000

OrEffFBSG = MaxOrFBRgPCSG / MaxOrFBSpBdSG * WgtFBOrRgSG Orthogonal Effectiveness F/B @ Start of Game 1.951 1.064 0.709 1.951 1.064 0.355 1.951 1.064 0.709 1.951 1.064 0.709 0.000 0.000 0.000 0.000 0.000 0.000

OrEffLRSG = MaxOrLRRgPCSG / MaxOrLRSpBdSG * WgtLROrRgSG Orthogonal Effectiveness L/R @ Start of Game 0.938 0.375 0.250 0.938 0.375 0.125 0.938 0.375 0.250 0.938 0.375 0.250 0.000 0.000 0.000 0.000 0.000 0.000

Dg Eff SG = MaxDgRgPcSG / MaxDgSpBdSG * WgtDgRgSG Diagonal Effectiveness @ Start of Game 1.309 0.714 0.238 1.309 0.714 0.238 0.000 0.000 0.000 0.000 0.000 0.000 1.309 0.714 0.357 1.309 0.714 0.357

L1Eff SG = MaxL1RgPcSG / MaxLSpBdSG * WgtL1L2CvSG L1 Effectiveness @ Start of Game 0.901 0.829 0.000 0.000 0.000 0.000 0.901 0.829 0.559 0.000 0.000 0.000 0.901 0.829 0.559 0.000 0.000 0.000

L2Eff SG = MaxL2RgPcSG / MaxLSpBdSG * WgtL1L2CvSG L2 Effectiveness @ Start of Game 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

EffRgSG = OrEffFBSG + OrEffLRSG + DgEffSg + L1EffSG + L2EffSG Effectiveness 5.098 2.982 1.197 4.197 2.153 0.718 3.789 2.268 1.518 2.888 1.439 0.959 2.210 1.543 0.916 1.309 0.714 0.357

FactorFWgt Factor F Weighting 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000

RgSG Range @ Start of Game

Factor F = EffRgSG * FactorFWgt Factor F 152.95 89.46 35.92 125.91 64.59 21.53 113.68 68.04 45.55 86.64 43.17 28.78 66.31 46.29 27.48 39.27 21.42 10.71

RgCB Range w/ Clear Board

MaxOrFBRgPcCB = MaxFBOrRgPc Maximum Orthogonal F/B Range of Piece w/ Clear Board 15.000 6.000 4.000 15.000 6.000 2.000 15.000 6.000 4.000 15.000 6.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxOrFBSpBdCB = RdFBSpBd Maximum Orthogonal F/B Span of Board w/ Clear Board 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

MaxOrLRRgPcCB =MaxLROrRgPc Maximum Orthogonal L/R Range of Piece w/ Clear Board 15.000 6.000 4.000 15.000 6.000 2.000 15.000 6.000 4.000 15.000 6.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxOrLRSpBdCB = CwLRSpBd Maximum Orthogonal L/R Span of Board w/ Clear Board 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

MaxDgRgPcCB = MaxDgRgPc Maximum Diagonal Range of Piece w/ Clear Board 14.500 6.000 2.000 14.500 6.000 2.000 0.000 0.000 0.000 0.000 0.000 0.000 14.500 6.000 3.000 14.500 6.000 3.000

MaxDgSpBdCB = Max (CwLRSpBd, RdFBSpBd ) Maximum Diagonal Span of Board w/ Clear Board 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

MaxL1RgPcCB = L1RgPc Maximum L1 Range of Piece w/ Clear Board 7.211 6.633 0.000 0.000 0.000 0.000 7.211 6.633 4.472 0.000 0.000 0.000 7.211 6.633 4.472 0.000 0.000 0.000

MaxL2RgPcCB = L2RgPc Maximum L2 Range of Piece w/ Clear Board 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

MaxLSpBdCB = Table G Maximum L Span of Board w/ Clear Board 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932

OrEffFBCB = MaxOrFBRgPcCB / MaxOrFBSpBdCB * WgtFBOrRgCB Orthogonal Effectiveness F/B w/ Clear Board 0.938 0.375 0.250 0.938 0.375 0.125 0.938 0.375 0.250 0.938 0.375 0.250 0.000 0.000 0.000 0.000 0.000 0.000

OrEffLRCB = MaxOrLRRgPcCB / MaxOrLRSpBdCB * WgtLROrRgCB Orthogonal Effectiveness L/R w/ Clear Board 0.938 0.375 0.250 0.938 0.375 0.125 0.938 0.375 0.250 0.938 0.375 0.250 0.000 0.000 0.000 0.000 0.000 0.000

DgEffCB = MaxDgRgPcCB / MaxDgSpBdCB * WgtDgCvCB Diagonal Effectiveness w/ Clear Board 0.906 0.375 0.125 0.906 0.375 0.125 0.000 0.000 0.000 0.000 0.000 0.000 0.906 0.375 0.188 0.906 0.375 0.188

L1EffCB = MaxL1RgPcCB / MaxLSpBdCB * WgtL1L2CvCB L1 Effectiveness w/ Clear Board 0.332 0.454 0.000 0.000 0.000 0.000 0.493 0.454 0.306 0.000 0.000 0.000 0.493 0.454 0.306 0.000 0.000 0.000

L2EffCB = MaxL2RgPcCB / MaxLSpBdCB * WgtL1L2CvCB L2 Effectiveness w/ Clear Board 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

EffRgCB = OrEffFBCB + OrEffLRCB +DgEffCB +L1EffCB + L2EffCB Effectiveness 3.113 1.579 0.625 2.781 1.125 0.375 2.368 1.204 0.806 1.875 0.750 0.500 1.399 0.829 0.493 0.906 0.375 0.188

FactorGWgt Factor G Weighting 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144

RgCB Range w/ Clear Board

Factor G Factor G 47.145 23.907 9.465 42.119 17.037 5.679 35.864 18.228 12.204 28.395 11.358 7.572 21.193 12.549 7.472 13.724 5.679 2.839

EXPANDED CHESS Appendix F

3 of 6

Weighting/Rating Summary Matrix

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

D E

Piece

TOTAL

Adjustment Factor

Piece Table Index

CwLRSpBd Columns wide L/R Span of Board

RdFBSpBd Rows deep F/B Span of Board

TotalSqBd = CwLRSpBd * RdFBSpBd Total Squares on Board

RwOcPrSG Rows Occupied/Player @ Start of Game

TotalOrCvPc Total Orthogonal Coverage of Piece

TotalDgCvPc Total Diagonal Coverage of Piece

L1CvPcSG L1 Coverage of Piece @ Start of Game

L2CvPcSG L2 Coverage of Piece @ Start of Game

L1CvPcCB L1 Coverage of Piece w/ Clear Board

L2CvPcCB L2 Coverage of Piece w/ Clear Board

L1RgPc L1 Range of Piece

L2RgPc L2 Range of Piece

MaxFBOrRgPc Maximum F/B Orthogonal Range of Piece

MaxLROrRgPc Maximum L/R Orthogonal Range of Piece

MaxDgRgPc Maximum Diagonal Range of Piece

WeightedOrRgPc Weighted Orthogonal Range of Piece

CvSG Coverage @ Start of Game

WgtOrCvSG Weighting Orthogonal Coverage

WgtDgCvSG Weighting Diagonal Coverage

WgtL1L2CvSG Weighting L1& L2 Coverage

CvCB Coverage w/ Clear Board

WgtOrCvCB Weighting Orthogonal Coverage

WgtDgCvCB Weighting Diagonal Coverage

WgtL1L2CvCB Weighting L1& L2 Coverage

RgSG Range @ Start of Game

WgtFBOrRgSG = WgtOrCvSG Weighting F/B Orthogonal Range

WgtLROrRgSG Weighting L/R Orthogonal Range

WgtDgRgSG Weighting Diagonal Range

WgtL1L2RgSG Weighting L1& L2 Range

RgCB Range w/ Clear Board

WgtFBOrRgCB Weighting F/B Orthogonal Range

WgtLROrRgCB Weighting L/R Orthogonal Range

WgtDgRgCB = WgtDgCvCB Weighting Diagonal Range

WgtL1L2RgCB Weighting L1& L2 Range

X Y Z AA AB AC AD AE AF AG AH AI AJ AK AL AM AN AO

Marshall Captain Sergeant Guard Archer Knight Catapult Ram Mace Lance Hatchet Sword Angel Saint Wizard Squire Knave Pawn

74.34 63.99 61.23 38.55 35.79 25.44 105.72 81.78 42.29 68.13 37.44 25.75 132.48 75.55 52.99 34.58 23.05 9.46

Marshall Captain Sergeant Guard Archer Knight Catapult Ram Mace Lance Hatchet Sword Angel Saint Wizard Squire Knave Pawn

7.856 6.763 6.471 4.074 3.782 2.689 11.173 8.643 4.469 7.200 3.956 2.721 14.001 7.984 5.600 3.654 2.436 1.000

Marshall Captain Sergeant Guard Archer Knight Catapult Ram Mace Lance Hatchet Sword Angel Saint Wizard Squire Knave Pawn

34.21 29.45 28.17 17.74 16.47 11.71 48.65 37.63 19.46 31.35 17.23 11.85 60.96 34.76 24.38 15.91 10.61 4.35

3 6 3 5 3 4 3 3 3 2 3 1 2 6 2 5 2 4 2 3 2 2 2 1 1 6 1 5 1 4 1 3 1 2 1 1

16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000

0.000 0.000 0.000 0.000 0.000 0.000 7.500 7.500 3.000 7.500 3.000 2.000 22.500 9.000 9.000 3.000 2.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 15.000 6.000 4.000 0.000 0.000 0.000 15.000 15.000 6.000 6.000 4.000 0.666

8.000 8.000 8.000 8.000 8.000 8.000 4.000 4.000 4.000 4.000 4.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

8.000 8.000 8.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

8.000 8.000 8.000 8.000 8.000 8.000 4.000 4.000 4.000 4.000 4.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

8.000 8.000 8.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

7.211 7.211 6.633 7.211 6.633 4.472 3.606 3.317 2.236 3.606 3.317 2.236 0.000 0.000 0.000 0.000 0.000 0.000

6.633 4.472 4.472 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 7.500 7.500 3.000 7.500 3.000 2.000 7.500 3.000 3.000 3.000 2.000 1.200

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 15.000 6.000 6.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 7.500 3.000 2.000 0.000 0.000 0.000 7.500 7.500 3.000 3.000 2.000 0.333

0.000 0.000 0.000 0.000 0.000 0.000 7.500 7.500 3.000 7.500 3.000 2.000 16.771 6.708 6.708 3.000 2.000 1.200

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643 1.643

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476 2.476

2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128 2.128

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428 1.428

2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500 2.500

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.500

EXPANDED CHESS Appendix F

4 of 6

Weighting/Rating Summary Matrix

4

D E

Piece

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

CvSG Coverage @ Start of Game

RwOcPrSG = RwOrPrSG Rows Occupied/Player @ Start of Game

PcPrSG = CwLRSpBd * RwOcPrSG Pieces/Player @ Start of Game

RwClrSG = RdFBSpBd * RwOcSG Rows Clear @ Start of Game

RwOcSG = RwOcPrSG * 2 Rows Occupied @ Start of Game

SqClrSG = RwOcPrSG * CwLRSpBd Squares Clear @ Start of Game

SqOcSG = RwOcSG * CwLRSpBd Squares Occupied @ Start of Game

SqOpSG = SClrSG - 1 Squares Open @ Start of Game

CwSG = CwLRSpBd Cw @ Start of Game

RdSG = RwClrSG Rd @ Start of Game

RdCwLTEQGTFlag (If CwSg < RdSg, 0,1) Rd/Cw LT/EQ-GT Flag

Rd EvenOddFlag (If RdSG = Ev, 0 / Od, 1) Rd Even/Odd Flag

DgFCv = Table A Diagonal Forward Coverage

DgBCv = Table B Diagonal Backward Coverage

DgTotalCv = DgFCv + DgBCv Diagonal Total Coverage

OrLRCv = CwSG - 1 Orthogonal L/R Coverage

OrFBCv = RdSG -1 Orthogonal F/B Coverage

OrTotalCv = OrLRCv + OrFBCv Orthogonal Total Coverage

MaxOrFBRgPcSG = Table C Maximum Orthogonal F/B Range of Piece @ Start of Game

MaxOrLRRgPcSG = MaxOrLRRg Maximum Orthogonal L/R Range of Piece @ Start of Game

MaxDgRgPcSG = Table D Maximum Diagonal Range of Piece @ Start of Game

OrSqCvPcSG = MaxOrFBRgPcSG + MaxOrLRRgPcSG Orthogonal Squares Covered by Piece @ Start of Game

DgSqCvPcSG = MaxDgRgPcSG * 2 Diagonal Squares Covered by Piece @ Start of Game

L1SqCvPcSG = L1CvPcSG L1 Squares Covered by Piece @ Start of Game

L2SqCvPcSG = L2CvPcSG L2 Squares Covered by Piece @ Start of Game

OrEffSG = OrSqCvPcSG/SqOpSG*WgtOrCvSG Orthogonal Effectiveness @ Start of Game

DgEffSG = DgSqCvPcSG/SqOpSG*WgtDgCvSg Diagonal Effectiveness @ Start of Game

L1Eff SG = L1SqCvPcSG / SqOPSG * WgtL1L2CvSG L1 Effectiveness @ Start of Game

L2EffSG = L2SqCvPcSG / SqOPSG * WgtL1L2CvSG L2 Effectiveness @ Start of Game

EffCvSG = OrEffSG + DgEffSg + L1EffSG + L2EffSG Effectiveness

FactorDWgt Factor D Weighting

CvSG Coverage @ Start of Game

FactorD = EffCvSG * FactorDWgt Factor D

CvCB Coverage w/ Clear Board

SqClrCB = TotalSqBd Squares Clear w/ Clear Board

SqOpenCB = SqClrCB - 1 Squares Open w/ Clear Board

OrSqCvCB = Total OrCvPc Orthogonal Squares Covered w/ Clear Board

DgSqCvCB = TotalDgCvPc Diagonal Squares Covered w/ Clear Board

L1SqCvCB = L1CvPcCB L1 Squares Covered w/ Clear Board

L2SqCvCB = L2CvPcCB L2 Squares Covered w Clear Board

OrEffCB = OrSqCvCB / SqOpenCB * WgtOrCvCB Orthogonal Efectiveness w/ Clear Board

DgEffCB = DGSqCvCB / SqOpenCB * WgtDgCvCB Diagonal Effectiveness w/ Clear Board

L1EffCB = L1SqCvCB / SqOpenCB * WgtL1L2CvCB L1 Effectiveness w/ Clear Board

L2EffCB = L2SqCvCB / SqOpenCB * WgtL1L2CvCB L2 Effectiveness w/ Clear Board

EffCvCB Effectiveness

FactorEWgt Factor E Weighting

CvCB Coverage w/ Clear Board

FactorE = EffCvCB * FactorEWgt Factor E

X Y Z AA AB AC AD AE AF AG AH AI AJ AK AL AM AN AO

Marshall Captain Sergeant Guard Archer Knight Catapult Ram Mace Lance Hatchet Sword Angel Saint Wizard Squire Knave Pawn

4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000

64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000

12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000 4.000

192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000 192.000

64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000 64.000

191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000 191.000

16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000

12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000

15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000 15.000

11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000 11.000

26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000 26.000

0.000 0.000 0.000 0.000 0.000 0.000 7.500 7.500 3.000 7.500 3.000 2.000 7.500 3.000 3.000 3.000 2.000 1.200

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 15.000 6.000 6.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 7.500 3.000 2.000 0.000 0.000 0.000 7.500 7.500 3.000 3.000 2.000 0.333

0.000 0.000 0.000 0.000 0.000 0.000 7.500 7.500 3.000 7.500 3.000 2.000 22.500 9.000 9.000 3.000 2.000 1.200

0.000 0.000 0.000 0.000 0.000 0.000 15.000 6.000 4.000 0.000 0.000 0.000 15.000 15.000 6.000 6.000 4.000 0.666

8.000 8.000 8.000 8.000 8.000 8.000 4.000 4.000 4.000 4.000 4.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

8.000 8.000 8.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.039 0.039 0.016 0.039 0.016 0.010 0.118 0.047 0.047 0.016 0.010 0.0063

0.000 0.000 0.000 0.000 0.000 0.000 0.079 0.031 0.021 0.000 0.000 0.000 0.079 0.079 0.031 0.031 0.021 0.0035

0.069 0.069 0.069 0.069 0.069 0.069 0.034 0.034 0.034 0.034 0.034 0.034 0.000 0.000 0.000 0.000 0.000 0.0000

0.069 0.069 0.069 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0000

0.138 0.138 0.138 0.069 0.069 0.069 0.152 0.105 0.071 0.074 0.050 0.045 0.196 0.126 0.079 0.047 0.031 0.0098

24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856 24.856

3.421 3.421 3.421 1.711 1.711 1.711 3.783 2.612 1.766 1.831 1.246 1.116 4.880 3.123 1.952 1.171 0.781 0.243

256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000 256.000

255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000 255.000

0.000 0.000 0.000 0.000 0.000 0.000 7.500 7.500 3.000 7.500 3.000 2.000 22.500 9.000 9.000 3.000 2.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 15.000 6.000 4.000 0.000 0.000 0.000 15.000 15.000 6.000 6.000 4.000 0.666

8.000 8.000 8.000 8.000 8.000 8.000 4.000 4.000 4.000 4.000 4.000 4.000 0.000 0.000 0.000 0.000 0.000 0.000

8.000 8.000 8.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.029 0.029 0.012 0.029 0.012 0.008 0.088 0.035 0.035 0.012 0.008 0.004

0.000 0.000 0.000 0.000 0.000 0.000 0.059 0.024 0.016 0.000 0.000 0.000 0.059 0.059 0.024 0.024 0.016 0.003

0.078 0.078 0.078 0.078 0.078 0.078 0.039 0.039 0.039 0.039 0.039 0.039 0.000 0.000 0.000 0.000 0.000 0.000

0.078 0.078 0.078 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.155 0.155 0.155 0.078 0.078 0.078 0.127 0.092 0.066 0.068 0.051 0.047 0.147 0.094 0.059 0.035 0.024 0.007

30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000

4.661 4.661 4.661 2.330 2.330 2.330 3.812 2.753 1.989 2.048 1.518 1.400 4.412 2.824 1.765 1.059 0.706 0.196

EXPANDED CHESS Appendix F

5 of 6

Weighting/Rating Summary Matrix

4

D E

Piece

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

RgSG Range @ Start of Game

MaxOrFBRgPcSG = MaxDgRgPcSG Maximum Orthogonal F/B Range of Piece @ Start of Game

MaxOrFBSpBdSG = RwClrSG Maximum Orthogonal F/B Span of Board @ Start of Game

MaxOrLRRgPcSG = MaxLROrRgPc Maximum Orthogonal L/R Range of Piece @ Start of Game

MaxOrLRSpBdSG = CwLRSpBd Maximum Orthogonal L/R Span of Board @ Start of Game

MaxDgRgPcSG = MaxDgRgPcSG Maximum Diagonal Range of Piece @ Start of Game

MaxDgSpBdSG = RwClrSG Maximum Diagonal Span of Board @ Start of Game

MaxL1RgPcSG = Table E Maximum L1 Range of Piece @ Start of Game

MaxL2RgPcSG = Table F Maximum L2 Range of Piece @ Start of Game

MaxLSpBdSG = Sqrt(MaxOrFBSpBdSg^2 + MaxOrLRSpBdSg^2) Maximum L Span of Board @ Start of Game

OrEffFBSG = MaxOrFBRgPCSG / MaxOrFBSpBdSG * WgtFBOrRgSG Orthogonal Effectiveness F/B @ Start of Game

OrEffLRSG = MaxOrLRRgPCSG / MaxOrLRSpBdSG * WgtLROrRgSG Orthogonal Effectiveness L/R @ Start of Game

Dg Eff SG = MaxDgRgPcSG / MaxDgSpBdSG * WgtDgRgSG Diagonal Effectiveness @ Start of Game

L1Eff SG = MaxL1RgPcSG / MaxLSpBdSG * WgtL1L2CvSG L1 Effectiveness @ Start of Game

L2Eff SG = MaxL2RgPcSG / MaxLSpBdSG * WgtL1L2CvSG L2 Effectiveness @ Start of Game

EffRgSG = OrEffFBSG + OrEffLRSG + DgEffSg + L1EffSG + L2EffSG Effectiveness

FactorFWgt Factor F Weighting

RgSG Range @ Start of Game

Factor F = EffRgSG * FactorFWgt Factor F

RgCB Range w/ Clear Board

MaxOrFBRgPcCB = MaxFBOrRgPc Maximum Orthogonal F/B Range of Piece w/ Clear Board

MaxOrFBSpBdCB = RdFBSpBd Maximum Orthogonal F/B Span of Board w/ Clear Board

MaxOrLRRgPcCB =MaxLROrRgPc Maximum Orthogonal L/R Range of Piece w/ Clear Board

MaxOrLRSpBdCB = CwLRSpBd Maximum Orthogonal L/R Span of Board w/ Clear Board

MaxDgRgPcCB = MaxDgRgPc Maximum Diagonal Range of Piece w/ Clear Board

MaxDgSpBdCB = Max (CwLRSpBd, RdFBSpBd ) Maximum Diagonal Span of Board w/ Clear Board

MaxL1RgPcCB = L1RgPc Maximum L1 Range of Piece w/ Clear Board

MaxL2RgPcCB = L2RgPc Maximum L2 Range of Piece w/ Clear Board

MaxLSpBdCB = Table G Maximum L Span of Board w/ Clear Board

OrEffFBCB = MaxOrFBRgPcCB / MaxOrFBSpBdCB * WgtFBOrRgCB Orthogonal Effectiveness F/B w/ Clear Board

OrEffLRCB = MaxOrLRRgPcCB / MaxOrLRSpBdCB * WgtLROrRgCB Orthogonal Effectiveness L/R w/ Clear Board

DgEffCB = MaxDgRgPcCB / MaxDgSpBdCB * WgtDgCvCB Diagonal Effectiveness w/ Clear Board

L1EffCB = MaxL1RgPcCB / MaxLSpBdCB * WgtL1L2CvCB L1 Effectiveness w/ Clear Board

L2EffCB = MaxL2RgPcCB / MaxLSpBdCB * WgtL1L2CvCB L2 Effectiveness w/ Clear Board

EffRgCB = OrEffFBCB + OrEffLRCB +DgEffCB +L1EffCB + L2EffCB Effectiveness

FactorGWgt Factor G Weighting

RgCB Range w/ Clear Board

Factor G Factor G

X Y Z AA AB AC AD AE AF AG AH AI AJ AK AL AM AN AO

Marshall Captain Sergeant Guard Archer Knight Catapult Ram Mace Lance Hatchet Sword Angel Saint Wizard Squire Knave Pawn

0.000 0.000 0.000 0.000 0.000 0.000 7.500 7.500 3.000 7.500 3.000 2.000 7.500 3.000 3.000 3.000 2.000 1.200

12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 15.000 6.000 6.000 0.000 0.000 0.000

16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

0.000 0.000 0.000 0.000 0.000 0.000 7.500 3.000 2.000 0.000 0.000 0.000 7.500 7.500 3.000 3.000 2.000 0.333

12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

7.211 7.211 6.633 7.211 6.633 4.472 3.606 3.317 2.236 3.606 3.317 2.236 0.000 0.000 0.000 0.000 0.000 0.000

6.633 4.472 4.472 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000 20.000

0.000 0.000 0.000 0.000 0.000 0.000 1.330 1.330 0.532 1.330 0.532 0.355 1.330 0.532 0.532 0.532 0.355 0.213

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.375 0.375 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.892 0.357 0.238 0.000 0.000 0.000 0.892 0.892 0.357 0.357 0.238 0.040

0.901 0.901 0.829 0.901 0.829 0.559 0.451 0.415 0.280 0.451 0.415 0.280 0.000 0.000 0.000 0.000 0.000 0.000

0.829 0.559 0.559 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1.731 1.460 1.388 0.901 0.829 0.559 2.673 2.102 1.049 1.781 0.947 0.634 3.160 1.799 1.264 0.889 0.593 0.252

30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000

51.92 43.81 41.65 27.04 24.87 16.77 80.19 63.05 31.48 53.42 28.40 19.02 94.80 53.98 37.92 26.67 17.78 7.57

0.000 0.000 0.000 0.000 0.000 0.000 7.500 7.500 3.000 7.500 3.000 2.000 7.500 3.000 3.000 3.000 2.000 1.200

16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 15.000 6.000 6.000 0.000 0.000 0.000

16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

0.000 0.000 0.000 0.000 0.000 0.000 7.500 3.000 2.000 0.000 0.000 0.000 7.500 7.500 3.000 3.000 2.000 0.333

16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000 16.000

7.211 7.211 6.633 7.211 6.633 4.472 3.606 3.317 2.236 3.606 3.317 2.236 0.000 0.000 0.000 0.000 0.000 0.000

6.633 4.472 4.472 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932 21.932

0.000 0.000 0.000 0.000 0.000 0.000 0.469 0.469 0.188 0.469 0.188 0.125 0.469 0.188 0.188 0.188 0.125 0.075

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.938 0.375 0.375 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.469 0.188 0.125 0.000 0.000 0.000 0.469 0.469 0.188 0.188 0.125 0.021

0.493 0.493 0.454 0.493 0.454 0.306 0.247 0.227 0.153 0.247 0.227 0.153 0.000 0.000 0.000 0.000 0.000 0.000

0.454 0.306 0.306 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.947 0.799 0.760 0.493 0.454 0.306 1.184 0.883 0.465 0.715 0.414 0.278 1.875 1.031 0.750 0.375 0.250 0.096

15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144 15.144

14.339 12.101 11.502 7.469 6.870 4.632 17.932 13.373 7.048 10.833 6.275 4.209 28.395 15.617 11.358 5.679 3.786 1.451

EXPANDED CHESS Appendix F

6 of 6