MAGIC SQUARE LEXICONmagic-squares.net/Downloads/HendricksBooks/Lexicon-v2.pdf · MAGIC SQUARE...

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H. D. Heinz & J. R. Hendricks

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All rows, columns, and 14 main diagonals sum correctly in proportion to length – 16 4x4 magic squares.

H. D. Heinz & J. R. Hendricks

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Magic Square Lexicon: Illustrated By

Harvey D. Heinz &

John R. Hendricks

Published November 2000 By HDH

Second print run July 2005

Copyright © 2000 by Harvey D. Heinz

Published in small quantities by HDH

as demand indicates

ISBN 0-9687985-0-0

Binding courtesy of Pacific Bindery Services Ltd.

For Erna & Celia Two ladies with patience and forbearance,

while their men are ‘playing with numbers’.

To commemorate the year 2000

Prime magic square A

67 241 577 571

547 769 127 13

223 139 421 673

619 307 331 199

Plus prime magic square B

1933 1759 1423 1429

1453 1231 1873 1987

1777 1861 1579 1327

1381 1693 1669 1801

Equals magic square C

2000 2000 2000 2000

2000 2000 2000 2000

2000 2000 2000 2000

2000 2000 2000 2000

Designed by John E. Everett (July, 2000) By permission, Carlos Rivera, http://www.primepuzzles.net/

Contents

Tables and Illustrations iii

Preface 1 xi

Preface 2 xiii

Magic Square Lexicon 1 to 174

References 175

The Authors 1 81

Magic Square Bibliography A1-1 to A1-15

John Hendricks Bibliography A2-1 to A2-3

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Tables and Illustrations

1 - Two types of almost magic stars ..................................................... 2

2 - The Alphamagic square of order-3 .................................................. 3

3 - An order-4 and an order-5 anti-magic square.................................. 4

4 - Number of anti-magic squares......................................................... 4

5 - Two order-5 anti-magic stars........................................................... 5

6 - The 8 aspects of a magic square. ..................................................... 6

7 - One of four order-3 magic cubes, all associated. ............................ 7

8 - Three associated magic squares....................................................... 8

9 - An order-9 associated, pandiagonal, 32 –ply magic square............. 9

10 - One of 4 basic magic cubes .........................................................11

11 - An order-4 Basic magic square ...................................................12

12 - A basic order-6 magic star...........................................................13

13 - Basic tesseract MT# 9 shown in the standard position................14

14 - A disguised version of MT# 9 (previous figure...........................15

15 - The 58 basic tesseracts of order-3 in indexed order ....................16

16 - An order-8 bent-diagonal magic square ......................................17

17 - An order-9 bimagic square with unusual features .......................19

18 - An order-5 and an order-6 bordered square.................................21

19 -The continuous nature of a pandiagonal magic square.................21

20 -Two order-3 magic squares and their complements .....................23

21 - An order-6 magic square and its complement .............................24

22 - An order-5 pandiagonal and its complement. pair pattern...........24

23 - An order-4 complete cube of binary digits ..................................25

24 - 12 order-3 magic squares form an order-12 composition ...........27

25 - Two order-4 magic squares formed from the above....................27

iii


26 - 2 order-5 magic square that don’t obey the rule for Bordered....28

27 - A hypercube of order-3 (a cube) showing the coordinates. .........30

28 - Crosmagic Quadrant pattern for order-9 and order-13 ................32

29 - Diamagic Quadrant pattern for order-9 and order-13. .................34

30 - Diametrically equidistant pairs in even and odd orders...............35

31 - A magic square in ternary, decimal 0-8 and decimal 1 to 9.........36

32 - Digital-root magic squares with digital roots of 3, 6 and 9. ........36

33 - Divide magic square from a multiply magic square. ...................37

34 - An order-4 domino magic square. ...............................................37

35 - An order-7 magic square using a complete set of dominoes. ......38

36 - Six doubly-even magic squares in one. ......................................39

37 - The 12 Dudeney groups. .............................................................40

38 - An order-7 magic square with embedded orders 3 and 4. ..........41

39 - Table – Summary of magic squares count...................................42

40 - Table – Summary of magic stars count. ......................................43

41 - Index # 6 of the 36 essentially different pandiagonals. ...............44

42 - An order-4, doubly-even, within a singly-even order-6. .............45

43 - An order-3 magic square with 2 expansion bands. ......................46

44 - Franklin’s order-8 magic square..................................................49

45 - Some Franklin order-8 patterns. ..................................................50

46 - Two magic Generalized parts. .....................................................52

47 - Exponential geometric magic square. .........................................53

48 - Ratio geometric magic square. ...................................................53

49 - Order-4 Latin, Greek and Graeco-Latin squares. ........................54

50 - A bipartite anti-magic graph........................................................54

iv


51 - A bipartite super-magic graph. ....................................................55

52 - An order-4 supermagic graph and magic square .........................55

53 - An order-5 star and two isomorphic tree-planting graphs. ..........56

54 - Orders 3 and 4 Heterosquare. ......................................................57

55 - Horizontal and vertical steps in an order-3 magic square............58

56 - This non-normal order-4 magic square does not use integers. ....59

57 - The first 3 order-4 magic squares. ...............................................60

58 - Star A shows the name of the cells, Star B is solution # 1. .........60

59 - The first six order-6 magic stars in tabular form. ........................61

60 - An order-5 with inlaid diamond and even corner numbers. ........62

61 - Comparison of order-4 Inlaid and Bordered magic squares. .......62

62 - The primary intermediate square for order-5...............................63

63 - Pandiagonal order-5 and solution set...........................................64

64 - An order-9 iso-like magic star. ....................................................66

65 - The order-9 diamagic square for the above star. .........................67

66 - An order-8 magic star isomorphic to an order-5 square. .............68

67 - Four aspects of the IXOHOXI magic square...............................69

68 - An order-8 with two half-board re-entrant Knight Tours. ...........71

69 - The two re-entrant knight tour paths for the above square. .........72

70 - Variations of a numerical Latin square........................................73

71 - Bergholt’s general form for order-4. ...........................................74

72 - A solution set and the resulting magic square. ............................75

73 - An order-7 Lozenge magic square. .............................................76

74 - Lringmagic Quadrant pattern for order-9 and order-13...............76

75 - An order-6 multiply magic square is 22 – ply and 32 – ply. .......77

v


76 - Two magic circles........................................................................78

77 - An order-3 basic normal magic cube...........................................79

78 - Number of lower hyperplanes within a given hypercube. ...........80

79 - An order-4 semi-pandiagonal and its magic line diagram. ..........81

80 - A 3 x 9 magic rectangle with correct diagonals...........................82

81 - An order-5 pandiagonal square with special numbers. ................83

82 - An order-9 pandiagonal magic square that is also 32-ply............83

83 - An order-12 pattern b normal magic star.....................................84

84 - An order-11 pattern b normal magic star.....................................85

85 - A summary of some magic star facts...........................................86

86 - Order-10 magic stars, Type S and Type T...................................87

87 - The 3-D order-3 magic star. ........................................................89

88 - Comparison of magic squares cubes and tesseracts. ...................90

89 - A magic tesseract shown with Hendricks projection...................91

90 - The above magic tesseract in tabular form. .................................92

91 - Order-3 hypercube comparison. ..................................................93

92 - Magic triangular regions, orders 4 and 6. ....................................94

93 - Transforming a 2-d magic star to 3-D cube and octahedron. ......95

94 - An order-4 magic square mapped to a tetrahedron.....................95

95 - An order-8 most-perfect magic square. .......................................97

96 - An order-4 multiplication magic square and its reverse. .............98

97 - # of segments in n-agonals for dimensions 2, 3, and 4 ................99

98 - Table - Hypercube has both a general and a specific meaning. 100

99 - Four different types of number squares .................................... 101

100 - This bordered magic square consists of two odd orders ......... 102

vi


101 - Two order-3 number squares and magic squares....................104

102 - An add magic square with an inlaid multiply magic square105

103 - A pentagram of five magic diamonds.....................................106

104 - This ornamental magic star consists of two interlocked stars.107

105 - An order-3 cube showing coordinates and line paths. ............108

106 - L. S. Frierson overlapping square, orders 3, 5, 7 and 9. .........109

107 - A. W. Johnson, Jr. bordered palindromic magic square. ........110

108 - An essentially different pandiagonal and a derivative. ...........112

109 - An order-7 pan-magic star. ....................................................113

110 - The order-7 pandiagonal square used for the above star. .......114

111 - A pan-3-agonal magic cube of order 4. ..................................116

112 - Order-5 pandiagonal magic square showing parity pattern. ...117

113 - An order-4 square partitioned into cells. ................................117

114 - Collison’s order-14 patchwork magic square. .......................118

115 - An Order-7 with inlaid order-5 pandiagonal magic square. ...120

116 - Values of the corner cells of the order-16 perfect tesseract...121

117 - Order-5 and order-11 perfect prime squares...........................123

118 - Two of 18 order-4 perimeter-magic triangles.........................124

119 - A perimeter-magic order-5 pentagon and order-3 septagon. 124

120 - Three of the 5 anti-magic Octahedrons. .................................125

121 - Plusmagic Quadrant pattern for order-9 and order-13............126

122 - Minimum starting prime for consecutive primes squares.......127

123 - A. W. Johnson, Jr.’s bordered, order-8 prime number square.127

124 - Two order-5 prime stars, minimal and consecutive primes. ...128

125 - Pythagorean magic squares, all order-4, Sc2 = sa2 + Sb2. ....130

vii


126 - Pythagorean magic squares, orders 3, 4, 5; Sc = sa + Sb.......130

127 - The eight opposite corner pairs of a magic tesseract. .............131

128 - Eight order-13 quadrant magic patterns ................................132.

129 - Even order crosmagic and lringmagic patterns, order 8, 12 ...133.

130 - This order-13 quadrant square is 14 times quadrant magic ....134.

131 - Some patterns of the order-13 quadrant magic square............135.

132 - A regular order-4 magic square from a Graeco-Latin square .137.

133 - Three representations of an order-4 magic square..................138.

134 - Continuous and separate patterns for Order-8 magic stars .....139.

135 - First four solutions, orders 8A and 8B....................................140.

136 - Two traditional tesseract projections. .....................................141

137 - The modern Hendrick’s projection.........................................141

138 - Order-4 reverse magic square pair..........................................142

139 - A principal reversible square with 2 of its 16 variations. .......143

140 - The 3 principal reversible squares of order-4. ........................144

141 - Number of Most-perfect magic squares .................................145

142 - The Sagrada magic square sums to 33....................................147

143 - The Sator word magic square. ................................................148

144 - An order-7 associated, and thus self-similar, magic square. ..149

145 - An order-4 self-similar magic square that is not associated. ..149

146 - An order-7 semi-magic square of squares. .............................150

147 - Two semi-pandiagonal magic squares. ...................................151

148 - An order-9 serrated magic square...........................................152

149 - Patterns for the above serrated magic square..........................153

150 - 2 pandiagonal squares contained in the above square. ...........153

viii


151 - An order-6 simple, normal, singly-even magic square. ..........155

152 - An algebraic pattern for an order-6 pandiagonal. ...................156

153 - The order-6 pandiagonal in base 7 and base-10 .....................157

154 - Even and odd number placement in a square and cube. .........158

155 - Species # 1 of the order-3 magic tesseract..............................159

156 - Square of primes make a semi-magic square..........................160

157 - The smallest orthomagic arrangement of distinct squares. .....161

158 - Sringmagic quadrant pattern for order-9 and order-13. ..........161

159 - Order-4, standard position, and two disguises. .......................162

160 - An order-6 magic star and a disguised version of it. .............163

161 - Subtraction magic square. ......................................................164

162 - Hypercubes – number of correct summations. .......................165

163 - Symmetrical cells in even and odd order magic squares. .......166

164 - Two Kravitz Talisman squares. ..............................................167

165 - Transforming an associated magic square to a pandiagonal...168

166 - # of parallel segmented triagonals for orders 3 to 10..............170

167 -Collison’s order-5 pandiagonal upside-down magic square. ...171

168 - The smallest possible consecutive primes order-3..................172

169 - The smallest possible consecutive primes type-2 order-3 ......172

170 - An order-8A weakly magic star..............................................173

171 - An unorthodox use of wrap-around........................................174

ix

7 47 11 40 6 49 15

48 31 18 13 12 45 8

4 26 33 30 41 14 27

16 29 22 25 28 21 34

23 36 9 20 17 24 46

42 5 38 37 32 19 2

35 1 44 10 39 3 43

PREFACE 1.

With the increasing popularity of the World Wide Web has come an explosive increase in published material on magic squares and cubes.

As I look at this material, I can appreciate how it is expanding our knowledge of this fascinating subject. However, frequently an author comes up with a new idea (or what he thinks is a new idea) and defines it using a term that has been in other use, in some cases, for hundreds of years.

On the other hand, because the subject is growing so fast, it is important that new words and phrases be defined and publicized as quickly as possible. For these reasons, in the winter of 1999 I decided to research this subject and publish a glossary on my Web page.

While this book is an attempt to standardize definitions, unfortunately not all magic square hobbyists will have a copy of this book at hand. Therefore, I suggest that when using a term not too well known, an attempt be made to clarify it’s meaning.

After posting the result to my site, I also printed it as a booklet for my personal reference. Because John Hendricks has been a good source of information for me, I sent him a copy of the booklet as a courtesy gesture. He then suggested I publish an expanded version of this in book form. I was immediately interested, and when he graciously accepted the request to serve as co-author, I decided, with his knowledge and experience to support me, I could do it.

What definitions have been included in this book is arbitrary. We have tried to include the more popular terms by drawing on a wide range of resources. Inevitably, with a book of this nature, personal preferences enter the picture. I am sure that every person reading this book will say to himself at some point why did he bother putting that item in, or why that illustration, or what about….

In any case I have worked on the assumption that a picture is worth a thousand words, and so have kept the descriptive text to a minimum. I have tried, when picking the illustrations, to find items of additional interest besides just referring to the particular term being defined. Hopefully, this will encourage the use of the book for browsing as well as for reference.

xi

Where I felt it would be appropriate, I have included a source reference. Where a definition appears or is used in a variety of sources, no mention is made of the source unless one particular location is especially informative. Where a term or definition is primarily (or solely) the work of one author, his work is cited as the source.

To add to the usefulness of this reference, I have in many cases, included relevant facts or tables of comparisons.

In the definitions text, bold type indicates a term that has its own definition.

This book uses m to indicate order (of magic squares, cubes, etc) and n to indicate dimension. This is the terminology used by Hendricks in his writings where so much of the work involves dimensions greater then 2. For magic stars, because all work is in two dimensions, the traditional n will continue to be used for the order.

Unless I specifically indicate otherwise, all references to magic squares mean normal (pure) magic squares composed of the natural numbers from 1 to m2. Likewise for cubes, tesseracts, etc. Normal magic stars use the numbers from 1 to 2n.

A special thanks to John Hendricks for the support and encouragement he has given me on this project.

Writing and publishing this book is a first venture for me. My hope is that it will prove to be an informative and a worthy reference on this fascinating subject.

Harvey D. Heinz

July 2005

It is now time to print a second run of this book. I have corrected the mistakes I am aware of that appeared in the first run.

The only other changes are minor variations in wording, or slight elaborations where space permitted.

H.D.H

xii

PREFACE 2.

“The analogy with squares and cubes is not complete, for rows of numbers can be arranged side-by-side to represent a visible square, squares can be piled one upon another to make a visible cube, but cubes cannot be so combined in drawing as to picture to the eye their higher relations.”

Magic Squares and Cubes, by W.S. Andrews, Dover Publication.

Faced with that, I proceeded anyway. Many professors, even today, teach the wrong model of the tesseract The problem has to do with partitioning the tesseract into cells, so that numbers can be assigned to various cells & coordinate positions.

In 1950, I sketched the first magic tesseract.. Nobody would look at it. Andrews had said it was impossible. I did not have a chance to look into it again for about five years and was on Gimli Airforce Station during a cold winter with not much else to do. So, I managed to make a 5- and 6-dimensional magic hypercube of order 3. I reasoned that if the establishment would not look at the magic tesseract, then they might look at the higher dimensional hypercubes. However, it was not until I was in Montreal before a mathematician from Seattle, home for Christmas, heard about me and wished to see my magic hypercubes. As he looked over it all, he said, “This stuff has got to be published.”

He phoned a friend at McGill University. The next thing I knew, I got a reprint order form for my article The Five- and Six-Dimensional Magic Hypercubes of Order 3 which was published in the Canadian Mathematical Bulletin, May 1962..

There were many hurdles to overcome with terminology and symbolism. A simple concept such as a row of numbers is not so simple in six-dimensional space. One runs out of names “row, column, pillar, post, file, rank,…then what?” So, the customary practice for higher dimensional spaces is to number the coordinate axes x1, x2, …xi…xn . Thus, I coined 1-row, 2-row, 3-row, ,,,, n-row.. There were both dimension and order to be taken into account now, so I used n for dimension and m for order. This was to be n-dimensional magic hypercubes of order m.

xiii

The old guard , adept in cubes, had just finished coining “long diagonal” and “space diagonal” and here I had to put a halt to that because for 4, 5, 6, 7, 8 dimensions you could not very well have “short diagonal”, “long diagonal” and “longer than that diagonal.” I noticed that when one traversed the square that 2 coordinates always changed on a diagonal as you moved along it. Three coordinates changed for the space diagonals of the cube, while only two coordinates changed for the facial diagonals. Therefore, it was clearly in order to talk in terms of 2-agonal, 3-agonal, 4-agonal, …, n-agonal depending upon how many coordinates change as you move along one of them. This means that triagonal, quadragonal, etc. were born and were a most logical solution to the problem..

One of the greatest challenges of all, was the concept of a “perfect cube.” As a boy, I learned that the four-space diagonals of a cube were required as well as all rows, columns and pillars to sum a constant magic sum. It was accepted that facial diagonals alone would be the requirement for a perfect cube. Eventually, Benson and Jacoby made a magic cube that had all broken triagonals and all broken diagonals summing the magic sum in every cross-section of the cube. It was both pandiagonal and pantriagonal. Thus, it was perfect.

Not until I made the perfect tesseract of order 16 and the 5-dimensional perfect magic hypercube of order 32 did I realize that perfect means all planar cross-sections are pandiagonal magic squares and all hypercubes have everything summing the magic sum

Planck had shown that the order of the hypercube had to be 2n or more before one could have pandiagonal squares with every cross-section. However, not until I actually made one did the point become clear. So the definition of “perfect” is upgraded. Through every cell on the hypercube there are (3n-1)/2 different routes that must sum the magic sum.

Over the years, it has been my pleasure to participate in the development of mathematics and to offer what I can on the subject.

John R. Hendricks

xiv

Algorithm … 1

AAAA Algorithm

A step-by step procedure for solving a problem by hand or by using a computer.

Algebraic pattern

A generalized magic square, cube, tesseract, hypercube, or border, etc. using algebraic digits for the numbers. A pattern is used extensively for making inlays. See Solution Set.

Almost-magic Stars

A magic pentagram (5-pointed star), we now know, must have 5 lines summing to an equal value. However, such a figure cannot be constructed using consecutive integers. Charles Trigg calls a pentagram with only 4 lines with equal sums but constructed with the consecutive numbers from 1 to 10, an almost-magic pentagram.

C. W. Trigg, J. Recreational Mathematics, 29:1, 1998, pp.8-11, Almost Magic

Pentagams

Marián Trenkler (Safarik University, Slovakia) has independently coined the phrase almost-magic, but generalizes it for all orders of stars. His definition: If there are numbers 1, 2, …, 2n located in a star Sn (or Tn) so that the sum on m – 2 lines is 4n + 2, on the others 4n + 1 and 4n + 3, we call it an almost-magic star. See Magic stars – type T for information on Sn and Tn.

Marián Trenkler, Magicke Hviezdy (Magic stars), Obsory Matematiky, Fyziky a

Informatiky, 51(1998).

2 … Magic Square Lexicon: Illustrated

….. Almost-magic Stars

NOTE that by Trenkler’s definition, the order-5 almost-magic star has only 3 lines summing correctly. Trigg’s order-5 (the only order he defines) requires 4 lines summing the same.

Neither author has defined almost-magic for higher order stars.

NOTE2: This book will retain the customary n as the order for magic stars but use m to indicate the order of magic squares, cubes, etc, leaving n free to indicate dimension.

1 – Two types of almost magic stars.

This Trigg Almost-magic order-5 star has 4 lines which sum to 24, and 1 line to 14. This Trenkler Almost-magic order-5 star has 3 lines which sum to 22, 1 line to 21 and 1 line to 23.

H.D.Heinz, http://www.geocities.com/~harveyh/trenkler.htm

C. W. Trigg, J. Recreational Mathematics, 29:1, 1998, pp.8-11, Almost Magic Pentagams

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4

9

105 3 6

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8

7

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910

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2 8

S5A

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6

3

Trigg Trenkler

Alphamagic square … 3

Alphamagic square

2 - The Alphamagic square of order-3.

Spell out the numbers in the first magic square. Count the letters in each number word and make a second magic square with these integers. Lee Sallows discovered this magic square oddity in 1986.

Lee Sallows, Abacus 4, 1986, pp28-45 & 1987 pp20-29

Anti-magic graphs

See Graphs – anti-magic

Anti-magic squares

An array of consecutive numbers, from 1 to m2, where the rows, columns and two main diagonals sum to a set of 2(m + 1) consecutive integers.

Anti-magic squares are a sub-set of heterosquares. Joseph S. Madachy, Mathemaics On Vacation, pp 101-110. (Also JRM 15:4, p.302)

5 22 18

28 15 2

12 8 25

4 9 8

11 7 3

6 5 10

Five twenty- two eighteen

Twenty- eight f if t een two

Twelve eight twenty- f ive


….. Anti-magic squares

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34 5 8 20 9 22 64

2 15 5 13 35 19 23 13 10 2 67

16 3 7 12 38 21 6 3 15 25 70

9 8 14 1 32 11 18 7 24 1 61

6 4 11 10 31 12 14 17 4 16 63

33 30 37 36 29 68 69 60 62 66 71

3 - An order-4 and an order-5 anti-magic square.

Note that in each case, the sums of the lines form a consecutive series.

In 1999, John Cormie, a graduate mathematics student at the University of Winnipeg, did a research project on this subject. He developed several methods of constructing these squares for both odd and even orders.

Order

(m)

Magic

squares

Anti-

magic

squares

1 1 0

2 0 0

3 1 0

4 880 299,710

5 275,305,224 ?

4 - Number of anti-magic squares. Cormie & Linek’s anti-magic square page is at

http://www.uwinnipeg.ca/~vlinek/jcormie/index.html

Anti-magic stars … 5

Anti-magic stars

A normal magic star diagram, but instead of each line of 4 numbers summing to a constant, each line has a different sum. If the sums consist of consecutive numbers, the star is anti-magic; if the sums are not consecutive, the star is a heterostar.

The illustration shows two of the 2208 possible order-5 anti-magic stars. Note that there can be no normal magic stars of order-5, that is those using the integers 1 to 10. The smallest series possible is 1 to 12 with no 7 or 11.

5 - Two order-5 anti-magic stars. C. Trigg, J. Recreational Mathematics, 10:3, 1977, pp 169-173, Anti-magic

pentagrams.

Arithmetic magic squares

Sometimes used to refer to squares that have a magic sum, especially to differentiate from geometric magic squares (Andrews).

Arrays

An array is an orderly arrangement of a set of cardinal numbers, algebraic symbols, or other elements into rows, columns, files, or any other lines.

1

72 4

5

8

10

3

9

6

1

7

24

5

8 10

39

6


….. Arrays

NOTE. For these purposes, the arrays used for magic squares, cubes and hypercubes would be narrowed down to square and rectangular ones like matrices and their cubic and higher dimensional equivalents. An array may also be a variable in a computer program. For these purposes, it would be the storage location for the magic square, cube, etc.

Aspect

An apparently different but in reality only a disguised version of the magic square, cube, tesseract, star, etc. It is obtained by rotations and/or reflections of the basic figure. Once one has a hypercube of any dimension, through mirror images and rotations one can view the hypercube in many ways. There are: A = (2n) n! ways of viewing a hypercube of dimension n.

Dimension (n) Name Aspects 2 square 8 3 cube 48 4 tesseract 384 5 hypercube 3840

In counting the number of any given type of hypercube, one can count all, including the aspects (the long count); or only the basic ones.

2 7 6 4 9 2 8 3 4 6 1 8

9 5 1 3 5 7 1 5 9 7 5 3

4 3 8 8 1 6 6 7 2 2 9 4

Original Rotate 90° Rotate 180° Rotate 270°

4 3 8 8 1 6 6 7 2 2 9 4

9 5 1 3 5 7 1 5 9 7 5 3

2 7 6 4 9 2 8 3 4 6 1 8

Vertical reflections of squares immediately above

6 - The 8 aspects of a magic square.

….. Aspect … 7

….. Aspect

Where n is the order (number of points) of a magic star there are 2n aspects for each star. NOTE that with rectilinear magic arrays, the number of aspects is determined by the dimension. With magic stars (which are normally only 2 dimensions) the number of aspects is determined by the order.

See Isomorphisms.

Associated magic cubes, tesseracts, etc.

Just as the one order-3 magic square is associated, so to are the 4 order-3 magic cubes and the 58 order-3 magic tesseracts. In fact, all order-3 magic hypercubes are associated.

All associated magic objects can be converted to another aspect by complementing each number (the self-similar feature). This figure is another aspect of the cube shown in basic magic cube.

7 - One of four order-3 magic cubes, all of which are associated.

Notice that the two numbers on each side of the center number sum to 28 which is 33 + 1. Go to Tesseract to see an order-3 associated 4-dimensional hypercube.

J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999, 0-9684700-0-9, p 59.

17

6

241

15 19 8

26 10

23 3 16

7 14 21

12 25 5

18 22 2

20 9 13

4 11 27


Associated magic square

A magic square where all pairs of cells diametrically equidistant from the center of the square equal the sum of the first and last terms of the

series, or m2 + 1 for a pure magic square. These number pairs are said to be complementary. This type of magic square is often referred to as a symmetrical magic square.

The center cell of odd order associated magic squares is always equal to the middle number of the series. Therefore the sum of each pair is equal to 2 times the center cell. In an order-5 magic square, the sum of the 2 symmetrical pairs plus the center cell is equal to the constant, and any two symmetrical pairs plus the center cell sum to the constant. i.e. the two pairs do not have to be symmetrical to each other.

In an even order magic square the sum of any m/2 symmetrical pairs will equal the constant (the sum of the 2 members of a symmetrical pair is equal to the sum of the first and last terms of the series).

8 - Three associated magic squares.

The order-3 associated magic square with each pair symmetrical summing to 32 + 1. The order-4 associated magic square with each pair symmetrical summing to 42 + 1 The order-5 associated magic square with each pair symmetrical summing to 52 + 1 .

As with any magic square, each associated magic square has 8 aspects due to rotations and reflections. any associated magic square can be converted to another aspect by complementing each number (the self-similar feature).

1 15 24 8 17

16 9 5 4 23 7 16 5 14

2 7 6 3 6 10 15 20 4 13 22 6

9 5 1 2 7 11 14 12 21 10 19 3

4 3 8 13 12 8 1 9 18 2 11 25

….. Associated magic square … 9

….. Associated magic square

There are NO singly-even Associated pure magic squares. The one order-3 magic square is associative. There are 48 order-4 associative magic squares.

Order-5 is the smallest order having associated magic squares that are also pandiagonal.

1 42 80 64 24 35 46 60 17

50 61 12 5 43 75 68 25 30

72 20 31 54 56 13 9 38 76

8 37 78 71 19 33 53 55 15

48 59 16 3 41 79 66 23 34

67 27 29 49 63 11 4 45 74

6 44 73 69 26 28 51 62 10

52 57 14 7 39 77 70 21 32

65 22 36 47 58 18 2 40 81

9 – An order-9 associated, pandiagonal, 32-ply magic square.

Associated magic squares are occasionally referred to as regular.

All associated magic squares are semi-pandiagonal but not all semi-pandiagonal magic squares are associated.

Note that while an associated magic square is also referred to as symmetrical, it should properly be called center symmetrical. There are magic squares (rare) that are symmetrical across a line.

W. S. Andrews, Magic squares & Cubes, 1917, p.266

Benson & Jacoby, Magic squares & Cubes, Dover 1976, 0-486-23236-0

Auxiliary square

See Intermediate square.


BBBB Base

Also called the radix. The number of distinct single-digit numbers, including zero in a counting system.

When the radix. exceeds ten, then more symbols then the familiar 0, 1, 2, 3,...,9 are required. Sometimes Greek symbols are used. More common is a, b, c, etc.

It is often convenient to use a number base equal to the order of the magic figure. The number of digits making up the number in each cell are then equal to the dimension of the magic figure. When the magic square (or other figure) is completely designed, the numbers are then converted to base 10 (decimal) and 1 is added to each to make the series range from 1 to mn. where m = the order and n = the dimension.

Basic magic cube

There are 4 basic magic cubes of order-3. All four are associated (as is the single basic magic square). The squares in the three center planes of these four cubes is magic. Each of the four may be disguised to make 48 other (apparently) different magic cubes by means of rotations and reflections. These variations are NOT normally considered as new cubes by the magic square researcher for the purposes of enumeration. They may become important to use in determining degree of rarity by a statistician. See Relative frequency.

Which of the 48 aspects is considered to be the basic cube? Normally that is of no importance. However, if listing all the magic cube solutions of a given order, it is necessary to have a standard position. The basic cube is determined in this case by three conditions.

• The bottom left corner is the smallest corner.

• The value of the second cell of the bottom row is smaller then the first cell of the second row.

• The value of the first cell of the second row is smaller then the second cell of the first pillar.

….. Basic magic cube … 11

….. Basic magic cube

This definition is modeled after Frénicle’s (1693) definition for the basic magic square, but with one important difference. Frénicle considered the starting point for the magic square to be the top left corner. We have set the starting point for the cube (and higher dimensions) to be the bottom left corner. This is consistent with the modern coordinate system in geometry.

While this term will not get the same frequency of use that the equivalent term for magic squares does, it is presented here in the interest of completeness.

10 - One of 4 basic magic cubes.

This diagram is the same magic cube illustrated in Fig. 7. However, this one is normalized to the basic position. The other is a disguised version of this. Point of interest. There are 58 basic magic tesseracts of order-3. Each may be disguised to make 384 other (apparently) different magic tesseracts by means of rotations and reflections. There is only 1 basic magic square of order-3.

See Aspects and Basic magic square.

14

17 6

24

1 15

19

8

26

10

7

21

12

25

5

18

22

2

20

9

13

4

11

27

16

3

23


Basic magic square

There is 1 basic magic square of order-3 and 880 of order-4, each with 7 variations due to rotations and reflections. These variations are called aspects or disguised versions.

In fact, any magic square may be disguised to make 7 other (apparently) different magic squares by means of rotations and reflections. These variations are NOT considered new magic squares for purposes of enumeration.

Any of the eight variations may be considered the basic one except for enumerating and listing them. Normally, which one you consider the basic one has no importance. However, for purposes of listing and counting, a standard must be defined. Refer to Aspect, Index and Standard Position for a more in-depth discussion of this subject. Basic magic squares are also known as Fundamental magic squares.

4 9 14 7

15 6 1 12

5 16 11 2

10 3 8 13

11 – An order-4 Basic magic square.

This order-4 is basic because

• The cell in the top left corner has the lowest value of any corner cell.

• The cell to the right of this corner cell has a lower value then the first cell of row two.

It is now possible to put this magic square in an ordered list where it appears as # 695 of 880.

Bensen & Jacoby, New Recreations with Magic Squares, Dover, 1976, 0-486-23236-0, p. 123

Basic magic star … 13

Basic magic star

All normal magic stars have n lines of 4 numbers that total to the magic sum. A magic star may be disguised to make 2n-1 apparently different magic stars where n is the order (number of points) of the magic star.

These variations are NOT considered new magic stars for purposes of enumeration. This is also referred to as a fundamental magic star. Any of these 2n variations may be considered the fundamental one. However, see Standard position, magic star and Index.

Figure 12 is a basic magic star because:

• The point cell with the smallest value is at the top.

• The value of the top right valley cell is lower then the top left one.

This star is number 31 in the indexed list of 80 order-6 basic magic stars.

Note: One of the authors (Heinz) has found all basic solutions for magic stars of order 5 to 11 (and some for higher orders).

12 - A basic order-6 magic star. H.D.Heinz, http://www.geocities.com/~harveyh/magicstar_def.htm

10

1

8

7

952

12

11 4

6

3


Basic magic tesseract

13 - Basic tesseract MT# 9 shown in the standard position.

One of the authors (Hendricks) has found and listed all 58 basic magic tesseracts of order-3. He lists and displays them in his book All Third-Order magic Tesseracts using the following indexing method:

• Identify the lowest of the 16 corner numbers.

• Take the adjacent number to this corner in each of the four lines. Rearrange these four numbers (if necessary) in ascending order and write them after the corner number.

In figure 13, the lowest corner number is 12 and the four numbers adjacent to it are 52, 61,62, and 76. Taking them in order; row, column, pillar and file they are already in ascending order, and, because the lowest corner is in the bottom left position we realize this tesseract is in the standard position. This definition is consistent with that of the Basic magic cube.

1276

35

522

69

5945

1948

471

16

6239

22

6138

2450

964

1481

2857

4026

7433

498

1178

3463

3723

5642

2554

168

1380

30

1873

3258

4421

476

70

66

65751

367710

297915

274155

67353

72546

1775

31

204360

row

column

pillar

file

….. Basic magic tesseract … 15

….. Basic magic tesseract

The following table is from page 3 of All Third Order Magic

Tesseracts. Column C shows the lowest corner number. Although 7. 8, and 9 sometimes serve as corners, they never serve as minimum corners.

In the table, Column # is for Magic Tesseract # (the order that Hendricks found the tesseract).

Column S is for species (based on arrangement of even and odd numbers).

Simply sorting the four numbers adjacent to the lowest corner insures that the tesseracts appear in index order (which is the order listed here).

For order-3 magic tesseracts, there are three species. For each of the 58 basic tesseracts, there are 384 aspects or disguised versions.

See Basic magic cube, Basic magic square, Index, Magic tesseract, Species and Standard position.

14 – A disguised version of MT# 9 (previous figure).

12

76

35

52

2

69

59

45

19

48

4

71

16

62

39

22

61

38

24

50

9

64

14

81

28

57

40

26

74

33

49

8

11

78

34

63

37

23

56

42

25

54

1

68

13

80

30

18

73

32

58

44

21

47

6

70

66

65

7

51

36

77

10

29

79

15

27

41

55

67

3

53

72

5

46

17

75

31

20

43

60


….. Basic magic tesseract

C Adjacent axis Numbers

# S C Adjacent axis Numbers

# S

1 45, 51, 53, 54 38 3 4 54, 66, 75, 80 53 3

1 45, 69, 71, 72 46 3 4 54, 72, 74, 75 55 3

1 51, 69, 77, 78 54 3 5 43, 46, 48, 54 51 3

1 53, 71, 77, 80 34 3 5 43, 64, 66, 72 43 3

1 54, 72, 78, 80 1 1 5 45, 46, 48, 52 45 3

2 43, 51, 52, 54 12 2 5 45, 64, 66, 70 39 3

2 43, 69, 70, 72 11 2 5 48, 72, 73, 79 19 2

2 45, 49. 52, 54 26 2 5 52, 72, 73, 75 21 2

2 46, 67, 70, 72 16 2 5 54, 66, 73, 79 20 2

2 49, 69, 76, 78 15 2 5 54, 70, 73, 75 6 2

2 51, 67, 76, 78 25 2 6 43, 46, 47, 52 28 2

2 52, 70, 78, 81 35 3 6 43, 64, 65, 70 29 2

2 52, 72, 76, 81 36 3 6 52, 70, 73, 74 10 3

2 52, 72, 78, 79 31 3 10 51, 59, 60, 78 30 2

2 54, 70, 76, 81 37 3 10 53, 59, 62, 80 27 2

2 54, 70, 78, 79 32 3 10 54, 60, 62, 81 49 3

2 54, 72, 76, 79 33 3 10 54, 60, 63, 80 47 3

3 43, 49, 52, 53 44 3 10 54, 62, 63, 78 48 3

3 43, 67, 70, 71 50 3 11 49, 58, 60, 78 42 3

3 52, 70, 76, 80 4 1 11 51, 58, 60, 76 40 3

3 53, 71, 77, 80 3 3 11 52, 61, 63, 78 5 2

4 45, 47, 48, 54 13 2 11 54, 60, 61, 79 7 2

4 45, 65, 66, 72 23 2 11 54, 61, 63, 76 17 2

4 47, 71, 74, 80 24 2 12 49, 58, 59, 76 22 2

4 48, 66, 80, 81 56 3 12 52, 61, 62, 76 9 3

4 48, 72, 74, 81 58 3 13 53, 56, 62, 74 41 3

4 48, 72, 75, 80 52 3 13 54, 57, 62, 75 2 2

4 53, 65, 74, 80 14 2 13 54, 57, 63, 74 18 2

4 54, 66, 74, 81 57 3 14 54, 57, 61, 73 8 3

15 - The 58 basic tesseracts of order-3 in indexed order (C = corner #, # = order discovered, S = species)

J. R. Hendricks, All Third Order Magic Tesseracts, self-published 1999,

0-9684700-2-5

Bent diagonals … 17

Bent diagonals

Diagonals that proceed only to the center of the magic square and then change direction by 90 degrees. For example, with an order-8 magic square, starting from the top left corner, one bent diagonal would consist of the first 4 cells down to the right, then the next 4 cells would go up to the right, ending in the top right corner. Bent diagonals are the prominent feature of Franklin magic squares (which are actually only semi-magic because the main diagonals do not sum correctly). Most bent-diagonal magic squares (and all order-4) have the bent-diagonals starting and ending only in the corners. However, some (including the order-8 example shown here) may use wrap-around but must be symmetric around either the horizontal or the vertical axis of the magic square.

For example: In the following magic square, line;

• 1 + 55 + 64 +10 + 47 + 25 + 18 + 40 is correct.

• 58 + 9 + 7 +52 + 21 + 34 + 48 + 31 is correct.

• 4 + 54+ 57 +15 + 42 + 32 + 19 + 37 is correct.

• 40 + 58 + 9 + 7 +10 + 24 + 39 + 41 is incorrect ( because it is not centered horizontally).

1 16 57 56 17 32 41 40

58 55 2 15 42 39 18 31

8 9 64 49 24 25 48 33

63 50 7 10 47 34 23 26

5 12 61 52 21 28 45 36

62 51 6 11 46 35 22 27

4 13 60 53 20 29 44 37

59 54 3 14 43 38 19 30

16 - An order-8 bent-diagonal magic square.

This remarkable bent-diagonal pandiagonal magic square has many combinations of 8 numbers that sum correctly to 260.


….. Bent diagonals

The Following are all magic:

• 8 Rows and 8 columns

• 16 Diagonals and broken diagonal pairs

• 8 Bent diagonals in each of 4 directions = 32 total

• Any 2 x 4 rectangle (including wrap-around)

• Any 2 x 2 square = 130 (including wrap-around)

• Corners of any 3 x 3, 4 x 4, 6 x 6 or 8 x 8 square = 130 (including wrap-around) This magic square is from David H. Ahl Computers in Mathematics: A Sourcebook

of Ideas. Creative Computer Press,1979, 0-916688-16-X, P. 117

Bimagic cube

A magic cube that is still magic when all integers contained within it are squared.

Hendricks announced the discovery of the worlds first bimagic cube on June 9, 2000. It is order 25 so consists of the first 253 natural numbers. The magic sum in each row, column, pillar, and the four main triagonals is 195,325. When each of the 15,625 numbers is squared, the magic sum is 2,034,700,525.

The numbers at the eight corners are; 3426, 14669, 6663, 14200, 9997, 5590,12584, and 4491.

J. R. Hendricks, A Bimagic Cube Order 25, self-published 1999, 0-9684700-6-8 and

& H. Danielsson, Printout of A Bimagic Cube Order 25, 2000

Bimagic square

If a certain magic square is still magic when each integer is raised to the second power, it is called bimagic. If (in addition to being bimagic) the integers in the square can be raised to the third power and the resulting square is still magic, the square is then called a trimagic square. These squares are also referred to as doublemagic and triplemagic. To date the smallest bimagic square seems to be order 8, and the smallest trimagic square order 32.

Benson & Jacoby, New Recreations in Magic Squares, Dover, 1976, 0-486-23236-0,

pp 78-92

….. Bimagic square … 19

….. Bimagic square

1 23 18 33 52 38 62 75 67

48 40 35 77 72 55 25 11 6

65 60 79 13 8 21 45 28 50

43 29 51 66 58 80 14 9 19

63 73 68 2 24 16 31 53 39

26 12 4 46 41 36 78 70 56

76 71 57 27 10 5 47 42 34

15 7 20 44 30 49 64 59 81

32 54 37 61 74 69 3 22 17

17 - An order-9 bimagic square with unusual features.

This special order-9 bimagic square was designed by John Hendricks in 1999. Each row, column, both diagonals and the 9 numbers in each 3 x 3 square sum to 369. If each of the 81 numbers are squared, the above combinations all sum to 20,049. Different versions of this bimagic square along with theory of construction appear in Bimagic Squares.

Aale de Winkel reports, based on John Hendricks digital equations, that there are 43,008 order-9 bimagic squares.

J. R. Hendricks, Bimagic Squares: Order 9 self-published 1999, 0-9684700-6-8

e-mail of May 14, 2000

Bordered magic square

It is possible to form a magic square (of any odd or even order) and then put a border of cells around it so that you get a new magic square of order m + 2 (and in fact keep doing this indefinitely). Each element of the inside magic square (order-3 or 4) must be increased by 2m + 2, with the remaining numbers (low and high) being placed in the border.

Or to put it differently, there must be (m2 -1)/2 lowest numbers

and their complements (the highest numbers) in the border where m

2 is the order of the square the border surrounds. This applies to each border.


….. Bordered magic square

The outside border is called the first border and the borders are numbered from the outside in.

When a border (or borders) is removed from a Bordered magic square, the square is still magic (although no longer normal). Any (or all) borders may be rotated and /or reflected and the square will still be magic. The Bordered Magic Square is similar but not identical to Concentric and Inlaid magic squares.

Orders 5 and 6 are the two smallest orders for which you can have a bordered magic square.

Benson & Jacoby, Magic squares & Cubes, Dover 1976, 0-486-23236-0, pp 26-33 W. S. Andrews, Magic squares & Cubes, 1917

There are 2880 basic order-5 bordered magic squares (not counting the 7 disguised versions of each). There are 328,458,240 different basic order-6 bordered magic squares. See Enumeration for more on this and order-6.

J. R. Hendricks, Magic Square Course, 1991, unpublished, pp 85-98

M. Kraitchik, Mathematical Recreations., Dover Publ. , 1942, 53-9354, pp 166-170

All bordered magic squares show a consistent relationship between the sum of the numbers in each border and the value of the center cell (or in the case of even order the sum of the center 4 cells. Notice that here we number the borders from the inside out.

For the order-3 square below: value of center cell = 13 sum of border 1 = 1 x 8 x 13 = 104 sum of border 2 = 2 x 8 x 13 = 208 next border if there was one would be 3 x 8 x center cell.

For the order-4 square below: value of center 4 cells = 74 sum of border 1 = 3 x 74 = 222 sum of border 2 = 5 x 74 = 370 next border if there was one would be 7 x sum of center 4 cells.

This feature also applies even if the number series is not consecutive, such as prime number magic squares

….. Bordered magic square … 21

….. Bordered magic square

1 34 33 32 9 2

3 4 18 21 19 29 11 18 20 25 8

25 12 11 16 1 30 22 23 13 16 7

6 17 13 9 20 6 17 12 26 19 31

24 10 15 14 2 10 24 21 15 14 27

7 22 8 5 23 35 3 4 5 28 36

18 - An order-5 and an order-6 bordered square.

Broken diagonal pair

Two short diagonals that are parallel to but on opposite sides of a main diagonal and together contain the same number of cells as are contained in each row, column and main diagonal (i.e. the order). These are sometimes referred to as pan-diagonals, and are the prominent feature of Pandiagonal magic squares.

J. L. Fults, Magic Squares, 1974

10 19 3 12 21 10 19 3 12 21 10 19 3 2 11 25 9 18 2 11 25 9 18 2 11 25

24 8 17 1 15 24 8 17 1 15 24 8 17

16 5 14 23 7 16 5 14 23 7 16 5 14

13 22 6 20 4 13 22 6 20 4 13 22 6

10 19 3 12 21 10 19 3 12 21 10 19 3

2 11 25 9 18 2 11 25 9 18 2 11 25

24 8 17 1 15 24 8 17 1 15 24 8 17 16 5 14 23 7 16 5 14 23 7 16 5 14 13 22 6 20 4 13 22 6 20 4 13 22 6

19 -The continuous nature of a pandiagonal magic square.

Notice how the two parts of the broken diagonal 24, 5, 6, 12, 18 of the center pandiagonal magic square may be considered joined to make a complete line of m (in this case 5) numbers.

See Modular space where the broken diagonals become continuous. See Pandiagonal, Pantriagonal, etc., for more on n-dimensional.


CCCC Cell

The basic element of a magic square, magic cube, magic star, etc. Each cell contains one number, usually an integer. However, it can hold a symbol or the coordinates of its location.

There are m2 cells in a magic square of order m, m3 cells in a magic cube, m4 cells in a magic tesseract, 2n cells in a normal magic star, etc. (Note the use of n for order of the magic star.)

RouseBall & Coxeter, Mathematical Recreations and Essays, 1892, 13 Edition,

p.194

Column

Each vertical sequence of numbers. There are m columns of height m in an order-m magic square.

See Orthogonals for a cube illustrating all the lines.

Complementary numbers

In a normal magic square, the first and last numbers in the series are complementary numbers. Their sum forms the next number in the series (m2 + 1). All other pairs of numbers which also sum to m2 + 1 are also complementary.

If the numbers are not consecutive (the magic square is not normal), the complement pair total is the sum of the first and the last number. Sets of two complementary numbers are sometimes called complementary pairs. Associated magic squares have the complementary pair numbers symmetrical around the center of the magic square.

Please see Associated and Complementary magic squares.

Following are complementary magic squares. Because they are associated, the middle number in the series is it’s own complement.

….. Complementary numbers … 23

….. Complementary numbers

8 1 6 1669 199 1249

3 5 7 619 1039 1459

4 9 2 829 1879 409

A B

2 9 4 409 1879 829

7 5 3 1459 1039 619

6 1 8 1249 199 1669

20 -Two order-3 magic squares and their complementary magic squares.

Each number in the bottom magic square is the complement of the number in the top magic square.

• Each pair sums to 10 which is 1+ 9 (the first and last numbers of the series). Also, because the series consist of 1 to m2 (this is a normal magic square), the sum is m2 + 1.

• Each pair in this prime number magic square sums to 2078 which is 199 + 1879 (the first and last numbers of the series).

Complementary magic squares

A well know method of transforming one magic square into another of the same order, is to simply complement each number. If the magic square is associated, the resulting square is self-similar. That is, it is the same as the original but rotated 90°. If the complement pairs are symmetrical across either the horizontal or vertical axis, the resulting complementary magic square is also self-similar but reflected horizontally or vertically respectively. Robert Sery refers to this process as Complementary Pair Interchange (CPI).

Complimenting works even if the numbers are not consecutive. See Complimentary numbers, figure 20B (above) and figure 21.


….. Complementary magic squares

1 35 36 7 29 42 49 15 14 43 21 8

47 17 12 45 19 10 3 33 38 5 31 40

6 30 41 2 34 37 44 20 9 48 16 13

43 21 8 49 15 14 7 29 42 1 35 36

5 31 40 3 33 38 45 19 10 47 17 12

48 16 13 44 20 9 2 34 37 6 30 41

21 - An order-6 magic square and its complementary.

An order-6 pandiagonal magic square using 36 of the numbers from1 to 49. (It is impossible to form an order-6 pandiagonal magic square using consecutive numbers.) It is transformed to another order-6 pandiagonal by subtracting each number from 50 (the sum of the first and last numbers).

Complementary pair patterns

The two numbers that together sum to the next number in the series are a complement pair. Join these two numbers with a line. The resulting pattern may be used as a method of classifying the magic squares of a given order. See Dudeney groups for more on this.

There is only 1 pattern for order-3 and 12 patterns for order-4. No one has yet figured out how many patterns there are for order-5 or higher.

22 - An order-5 pandiagonal magic square and its complementary pair pattern.

1 7 19 25 13

20 23 11 2 9

12 4 10 18 21

8 16 22 14 5

24 15 3 6 17

Complete projection cubes … 25

Complete projection cubes

On the sci.math newsgroup Dec. 2, 1996 K. S. Brown asked the following:

Do there exist 4x4x4 cubes of binary digits such that the “projection’ onto each face of the cube gives the decimal digits from 0 to 15 (binary 0000 to 1111)?

To state it differently; can each of the four rows of each of the four horizontal planes of an order-4 cube be filled with binary digits such that when read in either direction, the decimal integers 0 to 15 are obtained? And to make the cube complete, can this be done so the result is valid for each of the other two orthogonal sets of four planes?

The answer is yes! Dan Cass found such a cube (shown here), and posted it on Dec. 10, 1996

23 - An order-4 complete cube of binary digits.

K. S. Brown shows on his web site at http://www.seanet.com/~ksbrown/kmath353.htm that this solution is unique.

1 0

1 01

0

1

0

10

10

10

101 01010

10 1

01

0 10

10 10

10 10

1

0

10

101 0

10 10

10 1

0 101

0

10

10

10


Components

A magic square, or cube, may be broken down into parts which are called components. Some authors use the method of components to build their magic squares. One of these methods which is most meaningful is to show a magic square broken into two squares where the various digits are separated, as shown below:

4 9 2 1 2 0 0 2 1

3 5 7 + 0 1 2 + 2 1 0 + 1

8 1 6 2 0 1 1 0 2

Magic Square Decimal System

3’s digit ternary number system

Units digit ternary # system

where the first square is a magic square in the decimal number system. The second square is a Latin square in the ternary number system of the 3’s digit. The third square, is a Latin square in the ternary number system for the units digit and is a rotation of the second square, The number one at the end balances the equation.

Composition magic square

It is simple to construct magic squares of order mn (m times n) where m and n are themselves orders of magic squares. For a normal magic square of this type, the series used is from 1 to (mn)2. An order 9 composite magic square would consist of 9 order 3 magic squares themselves arranged as an order 3 magic square and using the series from 1 to 81. An order 12 composite magic square could be made from nine order 4 magic squares by arranging the order 4 squares themselves as an order-3 square, (or sixteen order 3 magic squares arranged as an order 4 magic square). In either case, the series used would be from 1 to 144.

The example order-12 composition magic square was constructed out of 16 order-3 magic squares. They are arranged as per the numbers in the order-4 pattern. Numbers used are consecutive from 1 to 144. The magic sums of these order-3 squares in turn form another order-4 magic square.

Composition magic square … 27

Composition magic square

8 1 6 125 118 123 62 55 60 107 100 105

3 5 7 120 122 124 57 59 61 102 104 106

4 9 2 121 126 119 58 63 56 103 108 101

134 127 132 35 28 33 80 73 78 53 46 51

129 131 133 30 32 34 75 77 79 48 50 52

130 135 128 31 36 29 76 81 74 49 54 47

89 82 87 44 37 42 143 136 141 26 19 24

84 86 88 39 41 43 138 140 142 21 23 25

85 90 83 40 45 38 139 144 137 22 27 20

71 64 69 98 91 96 17 10 15 116 109 114

66 68 70 93 95 97 12 14 16 111 113 115

67 72 65 94 99 92 13 18 11 112 117 110

24 - Twelve order-3 magic squares form an order-12 composition magic square with a magic sum of 870.

1 14 7 12 15 366 177 312

15 4 9 6 393 96 231 150

10 5 16 3 258 123 420 69

8 11 2 13 204 285 42 339

A. B.

25 - Two order-4 magic squares, from the order-12 composition magic square.

The order-4 pandiagonal magic square used as a pattern to place the order-3 squares.

The magic sum of each of the order-3 squares form an order 4 pandiagonal magic square with the magic sum 870.


Concentric magic square

Traditionally this has been another name for Bordered magic squares. It has also been used for Inlaid magic squares.

But Collison found several order-5 magic squares (2 are shown below) by computer search. They depend upon being able to carry over to the next column excess from the units column, which is not normally taken into account in constructing bordered or inlaid magic squares.

24 22 1 12 6 6 24 15 12 8

8 19 5 15 18 21 22 1 16 5

3 9 13 17 23 3 7 13 19 23

10 11 21 7 16 17 10 25 4 9

20 4 25 14 2 18 2 11 14 20

26 - Two order-5 magic square that don’t obey the rule for Bordered or Inlaid magic squares.

See Bordered and Inlaid magic squares. J. R. Hendricks, The Magic Square Course, self-published 1991, p 88

Congruence, Congruence equation.

See Modular Arithmetic.

Constant (S)

The sum produced by each row, column, and main diagonal (and possibly other arrangements). This value is also called the magic sum.

The constant (S) of a normal magic square is (m3+m)/2

If the magic square consists of consecutive numbers, but not starting at 1, the constant is (m3+m)/2+m(a-1) where a equals the starting number and m is the order. If the magic square consists of numbers with a fixed increment, then S = am + b(m/2)(m2-1) where a = starting number and b = increment. See Series.

….. Constant (S) … 29

….. Constant (S)

For a normal magic square, S = m(m2+1)/2.

For a normal magic cube, S = m(m3+1)/2.

For a tesseract S = m(m4+1)/2.

In general; for a n-dimensional hypercube S = m(mn+1)/2.

For a normal magic star, when n is the order, S = 4n + 2.

NOTE:Hendricks always uses m to indicate the order and reserves n to indicate the dimension of the magic object. See Magic sum and Summations for more information and comparison tables.

Continuous magic square

See Pandiagonal magic square.

Coordinates

A set of numbers that determine the location of a point (cell) in a space of a given dimension..

A coordinate system is normally not required for most work in magic squares. But, for 3-dimensions, or higher, a coordinate system is essential. Customarily, (x, y, z) are the coordinates for 3-dimensional space and (w, x, y, z) for 4-dimensional space.

Coordinates have been handled by Hendricks in a slightly different manner. For dimensions less then ten, only one digit is required per dimension, so the brackets and commas are not required, thus permitting a more concise and space saving notation.

For 2-dimensional space, the x-axis is in its customary position left-to-right. The y-axis is also in its usual position but is reversed. This is because of the way Frénicle defined the Basic magic square.

The origin is considered as being at the top-left, rather than the bottom left of the square.

Rows are parallel to the x-axis and columns are parallel to the y-axis. Pillars are parallel to the z-axis,


….. Coordinates

For three dimensions and higher a customary left-to-right x-axis; a front-to-back y-axis; and, a bottom to top z-axis is used. Then, the cube is ready to be presented in its usual presentation from the top layer down to the bottom layer. When working in 5- and 6-dimensional space and higher, it becomes more expedient to use numbered axes and the coordinates become:

(x1, x2, x3,…,xi,….., xn)

for an n-dimensional magic hypercube. All xi must lie between 1 and m inclusive which is the order of the hypercube. There really is no origin, or coordinate axes in Modular space. So, we simply define them as passing through the coordinate (1,1,1,…,1). A row would be parallel to the x1 axis, a column parallel to the x2 axis, a pillar parallel to the x3 axis , a file parallel to the x4 axis and we run out of names. Hence, we number the various kinds of rows according to which axis they are parallel to and say that an i-row lies parallel to the xi axis.

See modular space and orthogonal. See Journal of Recreational Mathematics, Vol.6, No. 3, 1973 pp.193-201. Magic

Tesseracts and n-Dimensional Magic Hypercubes.

Coordinates could also be considered as the indices (subscripts) of a variable array in a computer program used to store a magic square, cube, etc., being generated or displayed.

27 - A hypercube of order-3 showing the coordinates.

333

232

313

323

133

213

223

233

311

321

331

322

332

212

222

312

131

123

211

231

221

111

112

121

122

132

113

Coordinate iteration … 31

Coordinate iteration

Coordinate iteration is a systematic process of moving at unit intervals from 1 coordinate location to the next coordinate location along a line in modular space.

Moving along any orthogonal line requires changing only one coordinate digit, but moving along an n-agonal requires changing all n coordinate digits. See orthogonal for an illustration.

Coordinates could also be considered as the indices (subscripts) of a variable array in a computer program used to store a magic square, cube, etc., being generated or displayed. In this case, iterating one subscript at a time would permit storing (or retrieving) the value of the cells, as you move along the line.

See Pathfinder.

Corners

The corners are those cells where the lines that form the edges of the hypercube meet. They have coordinates which are either 1 or m, where m is the order of the hypercube. See Coordinates and Magic tesseract for illustrations.

There are 2n corners in a hypercube of dimension n.

Counting

How many magic squares, cubes, tesseracts, etc. are there?

There is a long count and a short count. Seasoned researchers in magic squares and cubes feel there is a duplication involved when you count rotations and reflections of a known square. Statisticians wishing to study the probability of a magic square, require to know them all.

The number of variations, called aspects, due to rotations and reflections varies with the dimension of the object. For a magic square (dimension 2) there are 8 aspects. So, for example, the researcher says there are 880 order-4 magic squares and the statistician claims there are 7040.Close attention must be paid to which number is being referred to. Normally, the count of magic squares considers the basic squares only.


….. Counting

Then there are unique magic squares that can be transformed to a range of magic squares. For example, order-5 has 3600 basic pandiagonal magic squares. They are derived from 36 essentially different squares that form 100 squares each by simple transformations. Furthermore, these 36 squares in turn can be formed from one square, using more complicated component substitution methods.

Bensen & Jacoby, New Recreations with Magic Squares, Dover, 1976, 0-486-23236-0, p. 125

For magic stars, the same consideration applies. However, here the number of aspects changes with the order of the star and is equal to 2n.

When talking about the number of magic objects, say magic squares, normally what is meant is the number of basic magic squares. However, keep the above considerations in mind and determine what is meant by the context.

Crosmagic

An array of m cells in the shape of an X that appears in each quadrant of an order-m quadrant magic square.

See Quadrant magic patterns and Quadrant magic square.

28 - Crosmagic Quadrant pattern for order-9 and order-13. H.D.Heinz, http://www.geocities.com/~harveyh/quadrant.htm

Cyclical magic squares

This is another, though seldom used, term for regular pandiagonal magic squares. See Regular and Irregular

Cyclical permutations … 33

Cyclical permutations

A pandiagonal magic square may be converted to another by simply moving one row or column to the opposite side of the square. For example, an order-5 pandiagonal magic square may be converted to 24 other pandiagonal magic squares. Any of the 25 numbers in the square may be brought to the top left corner (or any other position) by this method.

In 3-dimensional space, there can be cyclical permutations of a plane face of a cube to the other side of the cube. Pantriagonal magic cubes remain magic when this is done. In 4-dimensional space, entire cubes may be permuted. The Panquadragonal magic tesseract has this feature.

See also Transformations and Transposition.

J. R. Hendricks, American Mathematical Monthly, Vol. 75, No.4, p.384.

1

2

3

4

5 6

7

8 9

10

11

12

13


DDDD Degree (of a magic square.)

The power, or exponent to which the numbers must be raised, in order to achieve a magic square. The term is used in Bimagic and Trimagic squares.

Diabolic magic square

See Pandiagonal magic square.

Diagonal

Occasionally called a 2-agonal. See n-agonal. Also see Broken, Leading, Long, Main, Right, Opposite Short, Short.

Diagonal Latin square

A Latin square with the extra condition that both the diagonals also contain one of each symbol. See Latin square.

Diamagic

An array of m cells in the shape of a diamond that appears in each quadrant of an order-m quadrant magic square. For order-5 diamagic and crosmagic are the same.

29 - Diamagic Quadrant pattern for order-9 and order-13 H.D.Heinz, http://www.geocities.com/~harveyh/quadrant.htm

Diametrically equidistant … 35

Diametrically equidistant

A pair of cells the same distance from but on opposite sides of the center of the magic square. Other terms meaning the same thing are skew related and symmetrical cells. The two members of a complementary pair in an associated (symmetrical) magic square are diametrically equidistant.

X Y Z

Z Y

Y Z Z Y

X X

30 - Diametrically equidistant pairs X, Y and Z shown in an even and an odd order array.

Digital equations

One uses modular arithmetic in finding the various digits that comprise a number at a specific location in a magic square, or cube. If the digits of a number can be expressed as a function of their coordinate location, then the equation(s) describing the relationship can be called the digital equations. They are sometimes referred to as congruence equations or modular equations. For example: If at coordinate location (1, 3) we wish to find the number and it is known that:

D2 ≡ x + y (mod 3)

AndD1 ≡ 2x + y + 1 (mod 3

then the two digits D2 and D1 can be found.

D2 ≡ 1 + 3 ≡ 4 ≡ 1 (mod 3)

AndD1 ≡ 2 + 3 + 1 ≡ 6 ≡ 0 (mod 3)

So the number 10 is located at (1, 3).


….. Digital equations

10 is in the ternary number system because the modulus is 3. If the coordinate system is shown by:

(1,3), (2,3), (3,3) (1,2), (2,2), (3,2) (1,1), (2,1), (3,1)

and the numbers are all calculated as assigned to their respective locations, then one achieves the magic square below in the ternary number system which is then converted to decimal and finally 1 is added to each number.

10 22 01 3 8 1 4 9 2

02 11 20 2 4 6 3 5 7

21 00 12 7 0 5 8 1 6

31 - The magic square in ternary, decimal 0-8 and dec. 1 to 9. J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1998, 0-9684700-0-9

Digital-root magic squares

A digital root magic square is a number square consisting of sequential integers starting from 1 to m2 and with each line sum equal to the same digit when reduced to it’s digital root. This type of magic square was investigated by C. W. Trigg in 1984. He found that there are 27 basic squares of this type for order-3, nine each of digital root 3, 6, and 9.

1 3 8 3 4 8 5 1 3

2 4 6 1 5 9 4 6 8

9 5 7 2 6 7 9 2 7

32 - Digital-root magic squares with digital roots of 3, 6 and 9. C. W. Trigg, J. Recreational Mathematics, 17:2, 1978-79, pp.112-118, Nine-digit Digit-root Magic Squares.

Disguised magic square … 37

Disguised magic square

See Aspect and Basic magic square.

Division magic square

Construct an order-3 multiply magic square, then interchange diagonal opposite corners. Now, by multiplying the outside numbers of each line, and dividing by the middle number, the constant is obtained. See Geometric magic square for information and illustrations on multiply magic squares.

12 1 18 3 1 2

9 6 4 9 6 4

2 36 3 18 36 12

A B

33 - Divide magic square. A. multiply magic square, B. resulting divide magic square. RB = 5

Domino magic square

It is possible to arrange in the form of magic squares, any set of objects that contain number representations. Playing cards and dominos are two types that are often used. This order-4 requires duplicate dominoes and duplicate numbers but has four different numbers on each line.

34 – An order-4 Domino magic square.


….. Domino magic square

Here is an order-7 magic square that uses using a complete set of dominoes from double-0 to double-6.

35 - An order-7 magic square using a complete set of dominoes. RouseBall & Coxeter, Mathematical Recreations and Essays, 1892, 13 Edition,

p.214.

Double magic square

See bimagic magic square.

Doubly-even

The order (side) of the magic square is evenly divisible by 4. i.e. 4, 8, 12, etc. It is probably the easiest to construct.

The order-8 normal pandiagonal magic square shown here contains an order-4 pandiagonal (not normal) magic square in each quadrant and also an order-4 semi-pandiagonal magic square in the center.

….. Doubly-even … 39

….. Doubly-even

1 58 15 56 17 42 31 40

16 55 2 57 32 39 18 41

50 9 64 7 34 25 48 23

63 8 49 10 47 24 33 26

3 60 13 54 19 44 29 38

14 53 4 59 30 37 20 43

52 11 62 5 36 27 46 21

61 6 51 12 45 22 35 28

36 - Six doubly-even magic squares in one. J. R. Hendricks, The Magic Square Course, self-published 1991, p 205

Dudeney group patterns

When each pair of complementary numbers in a magic square are joined by a line, the resulting combination of lines forms a distinct pattern which may be called a complementary pair pattern.

H. E.Dudeney introduced this set of 12 patterns to classify the 880 order 4 magic squares. There are 48 group I, which are all pandiagonal. The 48 group III are associated All of groups II, III, IV and V are semi-pandiagonal, as are 96 of the 304 group VI. The other 448 order-4 magic squares are all simple.

Patterns I to III are fully symmetrical around the center point of the square. However, be aware that patterns IV to X also appear

rotated 90°for some of the basic magic squares. Pattern XI also

appears rotated 180° and 270° while pattern 12 appears rotated

90° and 180°.


….. Dudeney group patterns

37 - The 12 Dudeney groups.

H.E.Dudeney, Amusements in Mathematics, 1917, p 120

Jim Moran Magic Squares, 1981, 0-394-74798-4 (lots of material) Bensen & Jacoby, New Recreations with Magic Squares, 1976, 0-486-23236-0

I II III IV

V VI VII VIII

IX X XI XII

Embedded magic square … 41

E Embedded magic square

38 - An order-7 magic square with embedded orders 3 and 4.

A magic square (or other magic object) that has another magic square contained in it.

The order-7 is a simple, normal magic square. The order-4 (square cells) is simple and the order-3 (round cells) is associated. Neither of these two squares are normal. This square was designed by David Collison.

PS. There is a bonus ‘serrated’ magic diamond contained in this square! Can you see it? Rotate the square 45º to the right. The first 2 lines (of 6) of the diamond formation are 49, 1 and 7, 22,23, 48. Lines of two sum 50, lines of 3 = 75, of 4 = 100 and lines of 6 = 150.

J. R. Hendricks, The Magic Square Course, self-published 1991, pp 46-47

9 1 37 48 38 26 16

49 10 23 47 4 18 24

15 22 36 11 29 42 20

7 33 44 25 43 17 6

35 46 14 2 21 27 30

19 32 8 3 28 40 45

41 31 13 39 12 5 34


Enumeration – magic squares

This summary is concerned only with the basic number of each order. To determine the ‘long count’,. multiply the number shown here by the aspects. See Relative frequency.

Order Square Cube Tesser-

act

5-D 6-D

3 1 4 58 2992 543328

4 880 ? ? ? ?

5 275305224 ? ? ? ?

Aspects 8 48 384 3840 46080

39 - Table – Summary of magic squares count.

For Bordered magic squares of order-6 there are 81 different borders for the order-4 nucleus. There are 4! (factorial) different ways of permuting non-corner elements along a row edge. The same with a column edge. And there are 8 ways of rotating and reflecting the borders. So there are 81 x 4! x 4! x 8 borders and 880 basic order-4 nucleus. So there are 328,458,240 different order-6 basic bordered magic squares. For order-5, the total is much smaller because there are only 10 different borders, the variations are (3!)2 and there is only one nucleus order-3 magic square, making only 2880 basic bordered magic squares.

J. R. Hendricks, All Third Order Magic Tesseracts, self-published 1999, 0-9684700-2-5, The Magic Square Course, self-published 1991, pp. 85-98.

Enumeration – magic stars

This summary is concerned only with the basic number of each order. The basic count will always be an even number because all magic stars appear as a complement pair. To determine the ‘long count’. Multiply the number shown here by the aspects.

For magic stars, the order is the same as the number of points. Unlike magic squares, cubes, etc., regardless of the order of a normal magic star, there is always four numbers in a line. As the order increases, the variety of patterns increases. I have identified these patterns arbitrarily as A, B, C and D, which covers up to order-12.

….. Enumeration – magic stars … 43

….. Enumeration – magic stars

The total counts arrived at was obtained by exhaustive computer search. The validity of Heinz’s algorithm was confirmed by the counts for orders 6, 7 and 8 matching those obtained by other researchers in 1965. Also, all solutions found were successfully matched in complementary pairs.

Order Aspects A B C D

6 12 80 -- -- --

7 14 72 72 -- --

8 16 112 112 -- --

9 18 3014 1676 1676 --

10 20 115552 10882 10882 --

11 22 53528 75940 75940 53528

12 24 >500000 826112 >500000 >500000

40 - Table – Summary of magic stars count.

There are 12 basic magic stars of order-5 using integers from 1 to 12 with no 7 and 11. There are no normal magic stars of order-5. There are 168 Trigg type almost-magic stars (order-5) in 14 groups of 12. (Trigg). JRM 29(1), 1998 There are 2208 anti-magic stars order-5 (Trigg). JRM 10(3), 1977

Martin Gardner, Mathematical Carnival, Alfred A. Knoff, 1975, 0-394-49406-7 H.D.Heinz, http://www.geocities.com/~harveyh/magicstar_def.htm

Essentially different

There are 36 basic essentially different order-5 pandiagonal magic squares each of which have 99 variations by permutations of the rows, columns and diagonals. Rotations and reflections are not included in this count, so the total number of order-5 pandiagonal magic squares is 3600 time 8 rotations and reflections.. A magic square is essentially different from any other when it cannot be generated from another essentially different magic square by:

• any combination of the 1-3-5-2-4 transformation,

• the interchange of rows/columns with diagonals, and/or cyclical permutations.


….. Essentially different

Which of the set of 100 in each case is the essentially different one, is determined by

The number in the top left-hand corner is 1,

The number in the cell next to the 1 in the top row is less then any other number in the top row, in the left hand column or in the diagonal containing the 1, and

The number in the left-hand column of the second row is less then the number in the left-hand column of the last row.

There are 3 essentially different pandiagonal magic squares of order-4 each of which produces 16 variations..

There are 129,600 essentially different pandiagonal magic squares of order-7 each of which produces 294 variations (there are no normal pandiagonal magic squares of order-6).

1 7 14 20 23

15 18 21 2 9

22 4 10 13 16

8 11 17 24 5

19 25 3 6 12

41 - Index # 6 of the 36 essentially different pandiagonal magic squares of order-5. Bensen & Jacoby, New Recreations with Magic Squares, 1976, 0-486-23236-0,

p 129 & 139.

Eulerian square

See Graeco-Latin square.

Even-order … 45

Even-order

The order (side) of the magic square is evenly divisible by two.

See Doubly-even, Singly-even, Odd order.

This order-6 bordered magic square shown here is singly-even, the inside order-4 magic square is doubly-even.

4 27 8 31 32 9

2 11 25 14 24 35

7 22 16 19 17 30

34 23 13 26 12 3

36 18 20 15 21 1

28 10 29 6 5 33

42 - An order-4, which is doubly-even, within a singly-even order-6. J. R. Hendricks, The Magic Square Course, self-published 1991, p 32

Exhaustive search

How do you find all magic squares of a given order?

There are many different methods that may be used to generate magic squares. However, none will produce all magic squares. In this age of computers, the alternative is to do a search using an algorithm that is guaranteed to exhaust all possibilities i.e. to find all the possible magic squares (of the desired order).

The number of possibilities to be investigated grow very rapidly. Even lowly order-4 has 20,922,789,888,000 combinations of the numbers 1 to 16 (42 factorial). Because of this, it is a practical necessity to use shortcuts to eliminate impossible branches.

While all possible magic squares, including rotations and reflections will normally be found in an exhaustive search, it is simple to put conditions in the program to reject these disguised versions.


….. Exhaustive search

As a point of interest, a simple program Heinz wrote to produce a list of the 880 basic magic squares of order-4 took only 20 minutes on his 450 Mhz machine to search for and find all 880 basic solutions. It simply steps through the values for the variables A to P assigned to the cells in row order starting with the upper left cell. It has the following main features

A steps from 1 to 7 shortcut

B steps from 1 to 15

E steps from B + 1 to 16condition for basic magic square

D, M and P step from A + 1 to 16condition for basic square

Each line is tested when completed, program backtracks if line is not correct shortcut

First column is tested when completed, program backtracks if not correctshortcut

Expansion band

A band of cells that surround an inlaid or framed magic square. See Framed magic square.

26 3 21 33 45 15 32 18 34 10 19 30 38 26

38 27 14 37 8 39 12 47 27 2 41 44 11 3

30 2 22 49 4 48 20 29 14 4 43 28 36 21

19 41 7 25 43 9 31 17 37 49 25 1 13 33

10 44 46 1 28 6 40 5 8 22 7 46 42 45

34 11 36 13 42 23 16 35 39 48 9 6 23 15

18 47 29 17 5 35 24 24 16 40 31 20 12 32

A. B.

43 - An order-3 magic square with 2 expansion bands. S3 = 75, S5 = 125, S7 = 175.

….. Expansion band … 47

….. Expansion band

This illustration demonstrates a feature of framed magic squares (and bordered magic squares). Each band may be independently rotated and/or reflected. B. shows the inner order-3 (of fig. A) is

rotated 90° left, the first expansion band is reflected across the

lead diagonal and the outside band is rotated 90° right.

If used in an Inlaid magic cube, Hendricks refers to the expansion band as an expansion shell.

J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999, 0-9684700-1-7, pp35-48

87 80 85 57 24 17 22 57 69 62 67

82 84 86 57 19 21 23 57 64 66 68

83 88 81 57 20 25 18 57 65 70 63

57 57 57 57 57 57 57 57 57 57 57

42 35 40 57 60 53 58 57 78 71 76

37 39 41 57 55 57 59 57 73 75 77

38 43 36 57 56 61 54 57 74 79 72

57 57 57 57 57 57 57 57 57 57 57

51 44 49 57 96 89 94 57 33 26 31

46 48 50 57 91 93 95 57 28 30 32

47 52 45 57 92 97 90 57 29 34 27

48 … File

F File

The fourth dimension orthogonal line of numbers in a tesseract, or higher order hypercube. Analogous to rows and columns, the x and y direction lines of numbers in a magic square or cube, and pillars, the z direction in a magic cube.

J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1998, 0-9684700-0-9

Framed magic square

A subset of Inlaid magic square where an expansion band of numbers is placed around the inlaid magic square. Or the frame may be designed first, leaving room for the inlaid squares. The frame may be one, two, or even more rows and columns thick.

Framed and Bordered magic squares have many features in common. When a frame is removed from a Framed magic square, the square is still magic. Any (or all) frames may be rotated and /or reflected and the square will still be magic. There is a consistent relationship between the sum of all the cells in each expansion band and the center cell of the square.

However, unlike a Bordered magic square, the interior square may be a Normal magic square. It is not required that the (m2-1)/2 lowest and highest numbers of the series be in the expansion band. See Bordered magic square and Expansion bands.

J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999, 0-9684700-1-7

Franklin magic square … 49

Franklin magic square

This magic square was one of several created by Benjamin Franklin (1706 – 1790). It is not a true magic square because the main diagonals do not sum correctly. It is known mostly for it’s bent diagonals and the fact that he considered it the ‘most magical of all magic squares’.

It has many other patterns that are also magic as shown in the illustration below. Because the square is continuous (it wraps around), each pattern shown can start on any of the 64 squares.

Franklin created a similar one of order-16 that has even more patterns.

52 61 4 13 20 29 36 45

14 3 62 51 46 35 30 19

53 60 5 12 21 28 37 44

11 6 59 54 43 38 27 22

55 58 7 10 23 26 39 42

9 8 57 56 41 40 25 24

50 63 2 15 18 31 34 47

16 1 64 49 48 33 32 17

44 - Franklin’s order-8 magic square

On the next page as some of the patterns that are in this magic square. Each appears 64 times (using wrap-around). The 8 cell patterns all sum to 260, the 4 cell patterns to 130.

See also Bent diagonals


….. Franklin magic square

The bent diagonals

45 - Some of the Franklin order-8 patterns.

Fundamental Magic square, cube, tesseract, etc … 51

Fundamental Magic square, cube, tesseract, etc

The basic hypercube from which the other disguised versions are obtained. More commonly referred to as basic magic square, etc.

For more information see aspect, basic and standard position.

Fundamental magic star

The basic magic star from which 2n-1 other disguised versions are obtained. More commonly referred to as basic magic star.

For more information see aspect, basic and standard position.

1

2

11

9

12356

10

8

4

7

15

18

1314

25

21


GGGG

Generalized parts

‘Generalized Parts’ is a term coined by the late David Collison for formations used in his patchwork magic squares. The parts may be any size of magic square, rectangle, cross, elbow, tee, etc. The magic sum for each line is determined by the number of cells in the line.

The examples shown here use consecutive numbers (pure), but in context in a magic square, the numbers in the part will usually be non-consecutive.

1 12 1 16

11 2 14 3

5 4 10 7 4 9 7 5 11 15

9 8 3 6 13 8 12 10 6 2

Small elbow Tee

46- Two magic Generalized parts

See Patchwork magic squares for an example using large elbows and a cross.

J. R. Hendricks, Inlaid Magic Squares and Cubes, Self-published 1999,

0-9684700-1-7,Appendix

Geometric magic square

Instead of using numbers in arithmetic progression as in a Normal Magic Square , a geometric progression is used. These progressions may be exponential or ratio.

In the exponent type the numbers in the cells consist of a base value and an exponent. The base value is the same in each cell. The exponents are the numbers in a regular magic square.

Geometric magic square … 53

….. Geometric magic square

The ratio type uses a horizontal ratio and a vertical ratio. The constant is obtained by multiplying the cell contents. Geometric magic squares are the most common type of multiply magic squares.

210 2

1 2

7 1024 2 128

23 2

6 2

9 = 8 64 512

25 2

11 2

2 32 2048 4

A B

47 - Exponential geometric magic square.

A. Here a base of 2 is used. The exponents form a magic square with SA = 18.

B. The final Exponential geometric magic square with PB = 262,144. C.

1 2 4 50 1 20

5 10 20 4 10 25

25 50 100 5 100 2

A B

48 - Ratio geometric magic square.

A. Number square showing ratios; horizontal =2, vertical = 5. B. Final ratio geometric magic square, PB = 1000

W.S.Andrews, Magic Squares and Cubes, 1917, pp283-294.

Graeco -Latin square

When two Latin squares are constructed, one with Latin letters and one with Greek letters, in such a way that when superposed, each Latin letter appears once and only once with each Greek letter, the resulting square is called a Graeco-Latin square. This type of square is sometimes referred to as a Eulerian square. Instead of using Greek letters, it is more common nowadays to use upper and lower case Latin letters.

See Latin Square and Components.


….. Graeco -Latin square

a b c d αααα ββββ ΥΥΥΥ δδδδ aαaαaαaα bβbβbβbβ cΥcΥcΥcΥ dδdδdδdδ

c d a b δδδδ ΥΥΥΥ ββββ αααα cδcδcδcδ dΥdΥdΥdΥ aβaβaβaβ bαbαbαbα

d c b a ββββ αααα δδδδ ΥΥΥΥ dβdβdβdβ cαcαcαcα bδbδbδbδ aΥaΥaΥaΥ

b a d c ΥΥΥΥ δδδδ αααα ββββ bΥbΥbΥbΥ aδaδaδaδ dαdαdαdα ccccββββ

Latin square Greek square Graeco-Latin

49 - Order-4 Latin, Greek and Graeco-Latin squares.

Graph – anti-magic

A graph with q edges is said to be anti-magic if it is possible to label the edges with the numbers 1, 2, 3, …, q in such a way that at each vertex v the sum of the labels on the edges incident with v is different.

Many anti-magic graphs are isomorphic to magic squares, as the following example illustrates. This graph is isomorphic to the order-4 anti-magic square shown in Anti-magic squares. Solid vertices in this graph represent the rows of the magic square., hollow vertices the columns.

Note that unlike anti-magic squares, it is not required that anti-magic graphs have the sums form a consecutive series. In fact, for normal anti-magic squares, at least 1 of the two diagonals must sum to a value in the middle of the series. In the case of this graph, the sums form a series from 30 to 38 but with 34 missing.

50 – A bipartite antimagic graph. Hartsfield & Ringel, Journal of Recreational Mathematics, 21:2, 1989, pp107-115

1432 10

16

12

15 11

1

9

6 134 5

7 8

Graph – supermagic … 55

Graph – super-magic

A graph with q edges is said to be super-magic if it is possible to label the edges with the numbers 1, 2, 3, …, q in such a way that at each vertex v the sum of the labels on the edges incident with v is the same.

Many super-magic graphs are isomorphic to magic squares, as the following example illustrates. Solid vertices in this graph represent the rows of the magic square., hollow vertices the columns. This graph is bipartite because no two like vertices are directly connected by an edge.

51 – A bipartite super-magic graph. Hartsfield & Ringel, Journal of Recreational Mathematics, 21:2, 1989, pp107-115

52 – An order-4 bipartite super-magic graph.

51

3

4

5

6

78

921

2

34

67

8

9

1 16

2

3

4

5

6

78910

11

12

1314

15

1

1623

4

5

6 7

8

9

10 11

12

13

1415


Graphs – tree planting

A popular classification of recreational mathematics problems are known as tree-planting problems. The problem specifies how many trees, how many trees per row, and how many rows.

A magic star may be considered such a graph, with the trees represented by numbers. For order-5, there are six such graphs with five lines of four numbers and all being isomorphic.

Ten of the numbers 1 to 12 are used. Leaving out the 7 and the 11 gives 12 solutions with the 5 lines each summing to 24, the minimal solution. If the 2 and 8 are left out and the 7 and 11 used, there are 12 solutions with the magic sum of 28.

53 - An order-5 magic star and five isomorphic tree-planting graphs. H.D.Heinz, http://www.geocities.com/~harveyh/order5.htm

1

2

3

4

5

6

8

9

10

12

1

2

3 4

5

6

8 9

10 12 1

2

3

45

6

8

9

1012

1

2

34

5

6

89

10 12

1

2

3

4

5

6

8

9

10

12

1

23

45

6

8

9

10

12

Heterosquare … 57

HHHH Heterosquare

Similar to a magic square except all rows, columns, and main diagonals sum to different (not necessarily consecutive) integers. A simple method of generating any order heterosquare is to write the natural numbers from 1 to 9 in a spiral, starting from a corner and moving inward, or starting from the center and moving out. Another method that works for order-4 is to simply write the numbers in turn line by line, then interchange the 15 and 16 in the last two cells. A subset of heterosquare is the anti-magic square.

34

19 1 2 3 4 10

1 2 3 6 5 6 7 8 26

8 9 4 21 9 10 11 12 42

7 6 5 18 13 14 16 15 58

16 17 12 15 28 32 37 39 33

A. order-3, spirol B. Order-4, in line

54 - Orders 3 and 4 Heterosquares. Joseph S. Madachy, Mathemaics On Vacation, 1966, pp 101-110. (Also JRM 15:4, p.302)

Hexadecimant.

The 4-dimensional equivalent to the 2-dimensional quadrant and the 3-dimensional octant. These terms are generally more meaningful with magic squares, cubes and tesseracts of even order. A magic tesseract may be partitioned into 16 zones which are each called a hexadecimant.


Horizontal step

A magic square may consist of m series of numbers (m = order of the magic square).

This term refers to the difference between adjacent numbers in each of the m series. It is not a reference to the columns of the magic square. The difference between the last number of a series and the next number of the following series is called the vertical step.

In a normal magic square, the horizontal step and vertical step are both equal to 1.

1 2 3 10 1 7

5 6 7 3 6 9

9 10 11 5 11 2

A B

55 - Horizontal and vertical steps in a non-normal order-3 magic square.

See Order-3, Type 2, where the vertical step is a negative number. Vertical step also has more information and examples.

A is a number square showing horizontal step = 1, vertical step = 2; B is the resulting magic square.

If the numbers in each series are multiples of the first number in the series, the resulting square is multiply magic.


Horizontally paired

Two cells in the same row and the same distance from the center of the magic square.

Hypercube

A geometric figure consisting of all angles right and all sides equal. Normally applied to figures of five or more dimensions.

However, a square, cube and tesseract may be considered hypercubes of two, three and four dimensions. See magic

hypercube.

i-row … 59

IIII i-row

An i-row is a row, column, pillar, file, etc., of an n-dimensional hypercube of order-m. Some authors refer to these as “the orthogonals” because they are all mutually perpendicular to each other.

Customarily, a row runs from left to right; a column from front to back; a pillar runs up and down and a file runs obliquely to the other three in the projection of a tesseract. There are n(mn-1) i-rows in an n-dimensional hypercube of order-m. See orthogonals for an illustration.

Impure magic square

The numbers composing the magic square are not integers or are not in the range from 1 to m2.i.e. are not consecutive or the series does not start at 1. It may contain m series of m numbers where the horizontal and/or vertical steps are not 1, or it may contain numbers with random spacing between them.

447.25 545.25 538.25 468.25

524.25 482.25 489.25 503.25

496.25 510.25 517.25 475.25

531.25 461.25 454.25 552.25

56 - This non-normal order-4 magic square does not use integers.

This magic square with S = 1999 (for the year) is not normal, because the number series is not consecutive and it does not start with the number 1. Prime number magic squares are a class that obviously is not normal.

Indian Magic Square

See pandiagonal magic square


Index

The position in a list of magic squares of a given order where a given magic square fits, after it has been converted to the standard position. The correct placement or index of magic squares is determined by comparing each cell of two magic square of the same order starting with the top leftmost cell and proceeding across the top row, then across the second row, etc. until the two corresponding cells differ. The magic square with the smallest value in this cell is then given the lower index number. See also basic and normalized position. The concept of indexing is important because it permits direct comparison of lists of solutions compiled by different researchers. The index was designed by Bernard Frénicle de Bessy in 1693 when he published the 880 basic solutions for the order-4 magic square.

Magic stars may be indexed in a similar fashion.

Only normal magic squares and magic stars may be indexed because non-normal types of these cannot be ordered.

Index # 1 Index # 2 Index # 3

1 2 15 16 1 2 15 16 1 2 16 15

12 14 3 5 13 14 3 4 13 14 4 3

13 7 10 4 12 7 10 5 12 7 9 6

8 11 6 9 8 11 6 9 8 11 5 10

57 - The first 3 order-4 magic squares

58 - Star A shows the name of the cells, B is solution # 1.

A

B

C

DEFG

H

I J

K

L

1

2

3

4

56

7 89

10 11

12

A. B.

….. Index … 61

….. Index

# A b c D e f G h i J K L 1 1 2 11 12 3 5 6 10 9 8 4 7 2 1 2 11 12 4 3 7 8 10 5 6 9 3 1 2 12 11 3 4 8 7 10 5 6 9 4 1 2 12 11 4 5 6 10 9 7 3 8 5 1 3 10 12 2 4 8 6 11 5 9 7 6 1 3 10 12 2 7 5 9 11 8 6 4

79 5 3 7 11 1 4 10 2 9 6 12 8 80 6 1 9 10 2 3 11 4 5 8 7 12

59 - The first six and last two order-6 magic stars.

See Standard position, magic squares and Standard position, magic stars.

H.D.Heinz, http://www.geocities.com/~harveyh/magicstar_def.htm

Benson & Jacoby New Recreations with Magic Squares, 1976, p.123-124.

Inlaid magic cube.

A normal magic cube containing within itself a inlaid magic cube of lower order. There may also be or instead, inlaid magic squares within a cube. The world’s first magic cubes of this sort are in Hendricks’ book Inlaid Magic Squares and Cubes, 2

nd

edition.

J. R. Hendricks, Journal of Recreational Mathematics, 25:4, 1993, pp 286-288, An

Inlaid Magic Cube

Inlaid magic diamond

A magic diamond is a magic square rotated 45 degrees.

There are 8 possible basic order-5 magic squares that can contain an order-3 magic diamond. Each may be shown in 8 aspects due to rotations and/or reflections, making 64 in total.

The inlaid diamond may consist of all odd numbers or a mixture of odd and even numbers.


….. Inlaid magic diamond

2 10 19 14 20

22 3 21 11 8

17 25 13 1 9

18 15 5 23 4

6 12 7 16 24

60- An order-5 with inlaid diamond and even corner numbers.

This is one of two order-5 magic squares with inlaid diamonds that have even numbers at the corners (the other one is on the cover). The other six possible for this order have four odd numbers in the four corners and both even and odd numbers in the diamond.

J. R. Hendricks, Inlaid Magic Squares and Cubes, 1999, 0-9684700-1-7, pp 49-50.

Inlaid magic square

A magic square that contains within it other magic squares. However, unlike a bordered magic square, which must contain the lowest and highest numbers in the series, there is no such restriction on the inlaid magic square. It may even be a normal magic square. Inlays are often placed in the quadrants of a magic square, and the inlays may themselves contain inlays. Overlapping magic squares are a form of Inlaid and Patchwork magic squares. See also Concentric.

14 10 17 6 18 5 4 24 25 7

2 11 25 3 24 3 12 17 10 23

19 5 13 21 7 18 11 13 15 8

22 23 1 15 4 20 16 9 14 6

8 16 9 20 12 19 22 3 1 21

A. Inlaid B. Bordered

61 – Comparison of order-4 Inlaid and Bordered magic squares. J. R. Hendricks, The Magic Square Course, self-published 1991, p 32

J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999, 0-9684700-1-7

Inlaid magic tesseract … 63

Inlaid magic tesseract

A normal magic tesseract containing within itself an inlaid magic tesseract of lower order. There may also be, or instead, inlaid magic squares and/or cubes. The world’s first of this sort is a magic tesseract of order-6 with an inlaid magic tesseract of order-3, devised by Hendricks.

Hendricks supplies a copy of this tesseract with books ordered from him.

Intermediate square

An array formed with upper and lower case letters. By suitable arrangement of these letters and then assigning values to them, a magic square may be produced

For order-5, normally the values 0, 5, 10, 15 and 20 are assigned to the capital letters, and 1, 2, 3, 4 and 5 to the lower case letters in various combinations. If the letters are arranged so that one upper case and one lower case letter appears in each row, column and diagonal, the square is referred to as regular.

Note: Hendricks refers to this type of square as analytical. However, he uses base m, where m is the order or the magic square. The letters then represent the digits (i.e. the value are not added together). See representation.

Generally speaking, an intermediate square may be considered a square designed as an intermediate step in the construction of a magic square.

A + a B + b C + c D + d E + e

C + d D + e E + a A + b B + c

E + b A + c B + d C + e D + a

B + e C + a D + b E + c A + d

D + c E + d A + e B + a C + b

62 - The primary intermediate square for order-5.

All 3600 order-5 pandiagonal magic squares may be generated from this one basic intermediate square. Benson and Jacoby provide a concise table (see reference) to produce the 36 essentially different magic squares.


….. Intermediate square

Solution Set

1 7 13 19 25 A = 0 a = 1

14 20 21 2 8 B = 5 b = 2

22 3 9 15 16 C = 10 c = 3

10 11 17 23 4 D = 15 d = 4

18 24 5 6 12 E = 20 e = 5

63- Pandiagonal order-5 and solution set.

This is number 1 of the 36 essentially different pandiagonal squares of order-5. It is generated from the above intermediate square by assigning the values shown. See Solution set.

6 basic intermediate squares will generate all 38,102,400 regular pandiagonal magic squares of order-7.

W. Benson & O. Jacoby, New Recreations with Magic Squares, Dover Publ., 1976,

0-486-23236-0

Irregular

See Regular & Irregular

Iso-like magic star

An order-8A type magic star can be constructed by a systematic transformation of magic squares of certain orders. If the generating square is a plusmagic or diamagic square of order 8n+1 or 8n + 5 and the resulting magic star has 12 lines that sum correctly, it is an Iso-like magic star.

The name iso-like is derived from the fact that these stars are not quite isomorphic to the magic square because all the numbers contained in the square cannot be utilized in the star. See isomorphic magic stars and pan-magic stars which are closely related to Iso-like magic stars. The name pan-magic stars was coined by Aale de Winkel in 1999 when he started investigating their relationship with pandiagonal magic squares.

….. Iso-like magic star … 65

….. Iso-like magic star

One of us (Heinz) subsequently investigated the relationship to quadrant magic squares and redefined into three separate terms.

Comparison of features: Iso-like, Isomorphic and Pan-magic stars. In all cases the star is referred to as order-n where n is the order of the generating magic square.

Iso-like Generating magic square is quadrant magic and order 8n + 1 or 8n + 3. Resulting star is of pattern 8A with 12 lines of m numbers summing correctly. Magic square is plusmagic (required to form main diagonals). If magic square is diamagic, only 10 lines of m numbers sum correctly and it is called incomplete. No numbers are duplicated but not all numbers are used.

Isomorphic Generating magic square is order-4, any type, or order-5 plusmagic only. Order-4; the star will be type 8A normal with 8 lines of 4 numbers (mapping will vary). Order-5; the star will be type 8A with 12 lines of 5 numbers. In both cases, all numbers are used with no duplicates.

Pan-magic Generating magic square is pandiagonal odd order greater then five. Resulting star is of pattern 8A with 10 or 12 lines of m numbers summing correctly. Not all numbers are used and there will be either 4 or 8 numbers duplicated. A variation is the butterfly, a 12 pointed star with 20 lines of 9 numbers summing correctly. Note that pan-magic stars do not require that the generating square be quadrant magic, only pandiagonal.

Because this family of magic stars is always of pattern 8A, the order refers to the number of cells per line.



64 - An Order-9 incomplete iso-like magic star from an order-9 diamagic quadrant magic square.

This iso-like pattern has 10 lines of 9 numbers all summing correctly to 369. It is an order-8A star but in this case is referred to as order-9, the same as the generating square because all iso-like star patterns are the same, regardless of the order or the magic square. It is constructed from the order-9 diamagic magic square shown below and contains 65 numbers with no duplicates. If there are any plusmagic squares of order-9 (none have been found yet) a complete iso-like magic star (with 12 correct lines) can be obtained. Otherwise, the smallest complete iso-like star is be order-13.

35

5

50

57

20

36

44

78

64

16

49 39

85630

28

34

6

24

67

75

9

14 40

11

512779

69

7

54

10

58

19

61

73

15

81

41

1

71

12

74

17

252

66

45

80

72

22

59

29

42355

47

37

185226 68 31

2133

….. Iso-like magic star … 67


14 40 64 30 56 8 79 27 51

9 78 23 49 10 39 65 35 61

44 70 36 60 5 76 19 48 11

75 20 53 16 45 69 32 58 1

67 28 57 2 80 25 54 15 41

24 50 13 37 66 29 62 7 81

34 63 6 77 22 46 12 38 71

47 17 43 72 33 59 4 73 21

55 3 74 26 52 18 42 68 31

65 - The order-9 diamagic square used to generate the isolike magic star (above).

NOTE that we have used m to indicate the order of the magic squares but the traditional n for the order of the magic star.

See more on this subject at http://www.geocities.com/~harveyh/panmagic.htm

and http://www.adworks.myweb.nl/Magic/

Isomorphisms.

A one-to-one correspondence between the elements of two or more sets that preserves the structural properties of the domain. Aale de Winkel of the Netherlands who studies these things in n-dimensional space, points out that there are four basic isomorphisms which leave the numbers on the lines, (i-rows), parallel to the hypercube axes, merely reordered.

1. The reflection 2. The transposition: This is a reflection of a square across its main

diagonal. For a cube, this might be a rotation around its main triagonal.

3. The pan(re)location: For a magic square see Cyclical

Permutation. 4. The hyperagonal permutation.

See Aspects, Reflection, Rotation, Translocation.


Isomorphic magic stars

An order-8A type magic star constructed by a systematic transformation using all the numbers of an order 4 or 5 magic square.

If the magic square is order-4 then the resulting magic star is normal because it has 8 lines of 4 numbers that sum correctly.

If the originating magic square is order-5, it must be a plusmagic quadrant magic square and the resulting star has 12 lines of 5 numbers summing correctly. Because it contains five numbers per line, this star is not normal. Note that all pandiagonal magic squares are plusmagic quadrant magic squares but all plusmagic magic squares are not pandiagonal. In both cases all the numbers in the magic square are used to form the Isomorphic magic star.

See Iso-like magic stars and Pan-magic stars. Also my web page on Iso-like Magic Stars.

66 - An order-8A magic star isomorphic to an order-5 quadrant magic square. H.D.Heinz, http://www.geocities.com/~harveyh/panmagic.htm

1

19

23

15

7 25

9

17 1310

2

16

24

18

5

1220

8

214

6

22

14

3

11

1 7 14 25 18

15 23 16 2 9

17 4 10 13 21

8 11 22 19 5

24 20 3 6 12

Ixohoxi magic square … 69

Ixohoxi magic square

This novelty magic square is known as the IXOHOXI magic square. It is magic in all four of the above orientations. It is pan-diagonal so 4 rows, 4 columns, 2 main diagonals, 6 complementary diagonal pairs and 16 2 x 2 squares all sum to 19998.

Check this out with a mirror! All numbers in the reflection will read correct because both the one and the eight are symmetric about both the horizontal and the vertical axis. Note also that the name IXOHOXI has the same characteristics. See the Upside-down magic square which also relies on symmetrical digits 0, 6 and 9.

67 – Four aspects of the IXOHOXI magic square.

Original Rotated 180 degrees

Reflected verticalReflected horizontal


JJJJ Jaina magic square

Named for the first type of this square found as a Jaina inscription from the 12th or 13th century found in the City of Khajuraho, India.

Now commonly called pandiagonal magic squares.

W. S. Andrews, Magic Squares and Cubes, Dover, Publ., 1960, pp124-125

1

9

11

13

8

7

6

2

12

14

3

164 5

1015

18

27

292531

37 34

35

Knights tour magic square … 71

KKKK Knights tour magic square

The numbers are placed in the cells by following moves of a chess knight. Many such tours are possible and many of these symmetrically beautiful. If there is one knight move from the last move on the board back to the first move, the tour is said to be re-entrant. The problem is to end up with a square that is magic. It seems that no such magic square can exist for order-8, the size of a chess board. The best that can be hoped for is a semi-magic square with rows and columns, but not diagonals, summing correctly to 260.

However, re-entrant tour magic squares are possible for order-16 and magic squares of orders 4m greater then 16.

15 20 17 36 13 64 61 34

18 37 14 21 60 35 12 63

25 16 19 44 5 62 33 56

38 45 26 59 22 55 4 11

27 24 39 6 43 10 57 54

40 49 46 23 58 3 32 9

47 28 51 42 7 30 53 2

50 41 48 29 52 1 8 31

68 - An order-8 magic square with two half-board re-entrant Knight Tours.


….. Knights tour magic square

69 - The two re-entrant knight tour paths for the above magic square. RouseBall & Coxeter, Mathematical Recreations and Essays, 1892, 13 Edition,

1

2

3

4

5

6

7

8

9

10

11

12

1

2

3

4

5

6

7

8

9 10

11

12

1

2

3

4

5

6

7

8 9

10

11

12

1

2

3

4

5

6

7 8

9

10 11

12

S = 19

S = 20

S = 22

S = 21

Latin square …73

LLLL Latin square

A Latin square is an m x m array of m symbols in which each symbol appears exactly once in each row and each column of the array. It is not required that the same condition apply to the diagonal. If they do, the square is called a diagonal Latin square.

Latin squares are frequently used for generating magic squares. In this case, usually, but not always, they are diagonal Latin squares. The traditional literal symbols are used when algebraic digits are required. When numerical symbols are required, they

are specifically 0, 1, 2, …, m-1.

1 1 0 0

3 3 2 2

0 0 1 1

2 2 3 3

A simple number square but not Latin.

3 1 2 0

1 3 0 2

2 0 3 1

0 2 1 3

A bare Latin square which follows the definition.

1 0 3 2

3 2 1 0

0 1 2 3

2 3 0 1

A regular Latin square. The diagonals do not include 1 of each symbol, but do sum correctly

0 3 2 1

2 1 0 3

1 2 3 0

3 0 1 2

A diagonal or pure, Latin square. Both diagonals also contain one of each symbol.

70 - Variations of a numerical Latin square.


….. Latin square

The sum of a numerical Latin square is

See also Graeco-Latin squares.

J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999, 0-9684700-0-9, p4

Leading diagonal

Also called left diagonal. It is the line of numbers from the upper left corner of the magic square to the lower right corner. See Main Diagonals.

Lines of numbers

In a magic square, cube or hypercube these are more specifically referred to as rows, columns, diagonals, pillars, files, triagonals, quadragonals, etc. Each line contains m numbers where m is the order of the magic array. See also, orthogonals.

In a magic star they are the set of numbers forming a line between two points. In a normal magic star there is always four of these numbers per line, regardless of the order of the star. An ornamental magic star may have a set of any size.

Literal square

In 1910 Bergholt published a general form square that works with appropriate solution sets to generate any order-4 magic square.

A – a C + a + c B + b – c D - b

D + a – d B C A – a + d

C – b – d A D B + b – d

B + b D – a – c A – b + c C + a

71 - Bergholt’s general form for order-4

2

)1( −=

mmS

….. Literal square … 75

….. Literal square

A = 15 a = 12 3 10 16 5

B = 8 b = 4 13 8 2 11

C = 2 c = -4 6 15 9 4

D = 9 d = 8 12 1 7 14

72 - A solution set and the resulting magic square.

If a = b = d-c = (A – B – C + D)/2 the resulting square will be pandiagonal. If a + c = d = b – c and A + C = B + D the magic square will be associated. Therefore, the square cannot be both associated and pandiagonal because in that case A – a = B. In fact, the first associated, pandiagonal magic squares appear in order-5.

See Solution set.

RouseBall & Coxeter, Mathematical Recreations and Essays, 1892, 13 Edition, p.211

Long diagonal

Used by many authors on magic cubes to mean the diagonals which run from a corner of the cube, through the center to the opposite corner.

Hendricks uses triagonal, or 3-agonal instead.

See Triagonal and Short diagonal.

1

23

4

5

6

7

8

9

10

11

12

13

14

15 16

17

18 19

1

23

4

5

6

7

8

9

10

1112

13

14 15

16 17

18

19


Lozenge magic square

An odd order magic square where all the odd numbers are arranged sequentially to occupy a 45 degree rotated square in the center of the complete magic square. Unlike an Inlaid magic diamond, the lozenge (diamond) is not magic. The (m2

-1)/8 cells in each of the corner areas contain the even numbers.

The lozenge magic square is a prime example of a parity pattern because it’s main feature is the fact that the even and odd numbers are separated. See also Inlaid diamond.

18 10 2 43 42 34 26

12 4 45 37 29 28 20

6 47 39 31 23 15 14

49 41 33 25 17 9 1

36 35 27 19 11 3 44

30 22 21 13 5 46 38

24 16 8 7 48 40 32

73 - An order-7 Lozenge magic square.

Lringmagic

An array of n cells in the shape of a large ring that appears in each quadrant of an order-n quadrant magic square. For order-5 lringmagic and crosmagic are the same. See Quadrant magic patterns and Quadrant magic square.

74 - Lringmagic Quadrant pattern for order-9 and order-13 H.D.Heinz, http://www.geocities.com/~harveyh/quadrant.htm

M … 77

MMMM M

M indicates the order or number of cells per side of a magic hypercube of dimension n.

Many authors use n for this purpose. Hendricks (and this book) use m for this purpose and reserve n for specifying the dimension.

However, the more traditional n will be used to indicate the order of magic stars because normally they are only 2 dimensional (but see Magic star .. 3-D).

M2 – ply

A square is said to be m2 – ply when the number in any m x m group of cells give a constant sum in an arithmetic magic square , or a constant product in a geometric magic square.

The illustration shows an order-6 pandiagonal geometric magic square that is 22 – ply and 32 – ply. The magic product of any 2 x 2 square is 2,176,782,336 and any 3 x 3 squares is 1,023,490,369,077,469,249,536. The magic product of any of the 24 rows, columns or diagonals of the order-6 square is 101,559,956,668,416.

729 16 23328 1 11664 32

576 324 18 5184 36 162

3 3888 96 243 48 7776

46656 4 1458 64 2916 2

9 1296 288 81 144 2592

192 972 6 15552 12 486

75 - This order-6 pandiagonal multiply magic square is 22 – ply and 32 – ply. W. S. Andrews, Magic Squares and Cubes, Dover, Publ., 1960 ,p 292.


Magic circle, hexagon, cross, etc

Various arrangements of numbers, usually the first n integers, where all lines or points add up to the same constant value. See Generalized parts.

This magic circle demonstrates some of the characteristics of Order-4, Group II magic squares. The first figure has four numbers in the big circle, each of the four medium circles, and each of the five small circles, sum to 34. The second figure shows two other arrangements of four numbers. Note that the sixteen cells are arranged as a magic square.

76 - Two magic circles.

Magic cube

An m x m x m array of cells with each cell containing a number, usually an integer. These numbers are arranged so that the sum for each row, each column, each pillar, and the four main triagonals are all the same. Note that it is not required that the squares in the 3m planes of the cube have correct diagonals.

If the number series used is consecutive from 1 to m3., the cube is normal (see next entry). If you added 2 to each number of the cube in the next illustration, the result would be a non-normal (impure) magic cube Another example of an impure magic cube is one constructed only of prime numbers.

1

6

12

15

11

16

2

5

8

3

13

10

14

9

7

4

1

6

12

15

11

16

2

5

8

3

13

10

14

9

7

4

Magic cube, normal … 79

Magic cube, normal

Similar to a magic square but 3 dimensional instead of two. It contains the integers from 1 to m3. There are 3m2 + 4 lines that sum correctly. All rows, columns, pillars, and the four triagonals must sum to 1/2m(m3+1) (the constant). See Orthogonals. The minor diagonals do not sum correctly although it is possible that those in only one plane do.

There are 4 basic magic cubes of order-3, each of which can be shown in 48 aspects due to rotations and/or reflections. Following is one of them. See another at Basic magic cube….

Signatures for the four basic cubes are: 1, 15, 17, 23; 2, 15, 18, 24; 4, 17, 18, 26; 6, 16, 17, 26

77 - An order-3 basic normal magic cube.

Example lines: row; 4, 17, 21; column; 4, 18, 20; pillar; 4 26, 12; and triagonal; 4, 14, 24. All these lines must sum correctly for the cube to be magic. See Magic sum and Summations.

J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999

J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1998

Benson & Jacoby, Magic Cubes:New Recreations, 1981

14

17

6

24

1

15

19

8

26

10

7

21

12

25

518

22

2

20

9

13

4

11

27

16

3

23


Magic diamond

A magic diamond is a magic square rotated 45 degrees. It is quite similar to a Serrated magic square.

See Inlaid magic diamond.

Magic Graph

See Graphs

Magic hypercube

An n-dimensional array of mn cells containing the numbers1, 2,

…, mn arranged in such a way that all rows, columns, etc sum the

magic sum, as well as the 2n-1 n-agonals.

Dimension Hyper-planes contained in a hypercube

i-rows

Squares cubes Tesseracts 5-D Hyp.

2 2m 1 0 0 0

3 3m2 3m 1 0 0

4 4m3 6m2 4m 1 0

5 5m4 10m3 10m2 5m 1

6 6m5 15m4 20m3 15m2 6m

7 7m6 21m5 35m4 35m3 21m2

78 - Number of lower hyperplanes contained within a given hypercube.

The proceeding table shows the total number of hyper-planes in a hypercube. From this another table can be compiled that shows the number of bounding hyper-planes. Use m = 2 and change the word i-rows to edges. Remember that i-rows are orthogonals only. Correct n-agonals are not shown in this table. If the hypercube is perfect, all these hyper-planes must also have all the n-agonals summing correctly.

J. R. Hendricks, Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published,1999, 0-9684700-4-1, p.5.

Magic lines … 81

Magic lines

Lines connecting the centers of cells of a Pure Magic Square in the number order. The line diagrams produced may be used for purposes of classification.

If the areas between the lines are filled with contrasting colors, interesting abstract patterns result. These are also called sequence patterns.

79 - An order-4 semi-pandiagonal, its magic line diagram and the diagram filled in. Jim Moran, Magic Squares, 1981

See Dudeney patterns for another type of line pattern more commonly used for classification. It was first used by H.E. Dudeney to classify the 880 order 4 magic squares. In this method, each pair of complimentary numbers are joined by a line. The resulting combination of lines forms a distinct pattern.

H.E.Dudeney, Amusements in Mathematics, 1917, p 120

Jim Moran Magic Squares, 1981, 0-394-74798-4 (lots of material)

Magic object

Used in this book as a general term to indicate any array or other magic figure that has a magic sum, product, etc. that is in proportion to the number of cells in the line. Some magic objects are square, cube, star, circle, rectangle, generalized part, graph, etc.

141

2

8

3

4

5

6 7 9

16

1011

13

15

12


Magic rectangle

A rectangular array of cells numbered from 1 to m. All rows sum to the value which is the mean of each cell times the number of cells in the row. Likewise, all columns sum to the value which is the mean of each cell times the number of cells in the column. Neither Andrews, Collison, Hendricks, Moran, Trenkler, or de Winkel require that the diagonals be magic.

A simple way to construct a magic rectangle, is to take the several layer of a magic cube and place them side by side. If the cube is diagonal, pandiagonal or perfect, the diagonals will be correct, otherwise not.

Ed Shineman, in a letter dated March 27, 2000, provided a 4 x 16 magic rectangle in which 4 equally spaced leading and right diagonals summed correctly.

8 1 6 17 10 15 26 19 24

12 14 16 21 23 25 33 5 7

22 27 20 4 9 2 13 18 11

80 - A 3 x 9 magic rectangle with correct diagonals.

This 3 x 9 magic rectangle was constructed using a method proposed by Aale de Winkel. Columns and the 3 evenly spaced diagonals in each direction (starting at columns 1, 4 and 7) sum to 42, rows sum to 126.

e-mail from Aale de Winkel May 16, 2000

Magic square

An m x m array of cells with each cell containing a number. These numbers are arranged so that the sum for each row, each column, and the two main diagonals are all the same. If the numbers used are from 1 to m2 it is a normal magic square. If it has no special features, it is a simple magic square.

n is called the order by many authors so in that case the array is n x n.

NOTE: In geometry, n is used for the dimension, so when Hendricks extended the notion of magic squares to higher dimensions, he found it more practical to use m for the order and reserve n for the dimension of the magic hypercube. This is the practice that will be followed throughout this book.

….. Magic square … 83

….. Magic square

32 4 23 3 28

17 12 49 5 7

22 8 1 26 33

10 47 6 25 2

9 19 11 31 20

81 - An order-5 pandiagonal square with magic sum = 90 and containing special numbers.

This magic square was designed to celebrate one of the authors (Heinz) fathers 90th birthday. In the top row are 3 numbers of his birth date, April 23, 1903. Because the numbers in this square are not consecutive and starting with 1, this is not a normal magic square. However, because all diagonals sum correctly, it is not simple.

Magic square, normal

A magic square composed of the natural numbers from 1 to m2. Also called pure, or traditional.

64 24 35 46 60 17 1 42 80

52 57 14 7 39 77 70 21 32

9 40 74 72 22 29 54 58 11

69 26 28 51 62 10 6 44 73

48 59 16 3 41 79 66 23 34

4 38 81 67 20 36 49 56 18

71 19 33 53 55 15 8 37 78

50 61 12 5 43 75 68 25 30

2 45 76 65 27 31 47 63 13

82 - An order-9 pandiagonal magic square that is also 32-ply.

The equation for the constant of a magic square is S = (m3 + m)/2


….. Magic square, normal

Figure 82 is an normal magic square because it uses the numbers from 1 to m2 (1 –81). It is a perfect (pandiagonal) magic square because all 2m diagonals sum correctly. And it is 32 – ply because all 3 x 3 sub-squares sum correctly.

Magic star

The magic star shown here is index number 437363 of a total of 826112 basic order-12 magic stars of pattern B. Each of 12 lines of 4 numbers sum to 50. The complement of this star is # 3737 and the complement pair is # 1960. It is normal because it uses the consecutive numbers from `1 to 2n.

There are a total of 4 patterns for this order. The 826112 solutions of pattern b were found by an exhaustive computer search lasting 39.5 days. I have found many solutions for each of the other order-12 patterns, but have been reluctant to devote the necessary computer time to find all of them.

Note the use of n to indicate order. For orthogonal magic arrays, this book uses m to indicate order, and n to indicate the dimension. See Magic star, normal for more information on this subject and an order-11 example.

83 - An order-12 pattern b normal magic star.

12b

120

19

10

14

21

517

24

4

23

12

13

11

8

16

2

3

18

622

9

15 7

Index # 437363

This is the first basic star solution in the indexed list with 2 as the top point number. See Index.

H.D.Heinz,

http://www.geocities.com/~harveyh/magicstar.htm

Magic star, normal … 85

Magic star, normal

A normal magic star consists of a set of integers 1, 2, 3, ..., 2n which are placed at the 2n exterior points of intersection of the lines which form a regular polygram, such that the sum of the four integers found in any of the n lines is given by: S = 4n+2 where S is called the magic sum, and n is the order of the star. Also called pure.

Trenkler calls stars that do not consist of consecutive numbers starting with 1, Weakly magic stars, and those with four numbers per line but the two inside numbers placed at interior line intersections, Type-T stars.

Magic stars have not attracted the attention of mathematicians to the same extent that magic squares have. Presumably this is because the unstructured nature of magic stars do not lend themselves to mathematical analysis as magic squares do. These solutions were all found by exhaustive computer search.

84 - An order-11 pattern b normal magic star.

This is the first basic star solution in the indexed list with 12 as the top point number. See Index.

Note: All order 5 magic stars are not normal because it is impossible to construct a magic star of this order using consecutive numbers.

11B3

1

22

202

8

16

9

6

15

17

13

10

21

4

12

14

11

18

7

5

19

11bIndex # 75931


….. Magic Star, Normal

Order #

# Series

S Patterns (graphs)

Basic Solutions

Aspects

5 1 to 12 24 Continuous 12* 10

6 1 to 12 26 2 triangles 80 12

7 1 to 14 30 a-Continuous b-Continuous

72 72

14

8 1 to 16 34 a-2 squares b-Continuous

112 112

16

9 1 to 18 38 a-Continuous b-3 triangles c- Continuous

3014 1676 1676

18

10 1 to 20 42 a-2 pentagrams b-Continuous c-2 pentagons

10882 115522 10882

20

11 to 22 46 a-Continuous b-Continuous d-Continuous c-Continuous

53528 75940 53528 75940

22

12 1 to 24 50 a-2 hexagons b-3 squares c-4 triangles d- Continuous

>600000 826112 >600000 >600000

24

85 - A summary of some magic star facts.

Three of the four order-12 lists have not been completed. The large number of order-10b solutions looks suspicious. However, all solutions form complement pairs and I have not been able to find any duplicates. Order-5 is not normal because 7 and 11 are not used.

The number of aspects for each pattern of each order is equal to the number of points (rotation) times 2 (reflection).

H.D.Heinz, http://www.geocities.com/~harveyh/magicstar.htm

Magic stars – type T …87

Magic stars – type T

Marián Trenkler of Safarik University, Slovakia, has classified normal magic stars into two groups. Type S which is the same as my definition of a normal magic star, contains all the cells in the outside vertices of the star. The other is type T, which has two of the four cells of each line in the interior of the star.

86 - Order-10 magic stars, Type S and Type T.

The type S is a normal magic square by one of the authors (Heinz) definition and is index # 12195 of pattern B. The type T is not normal by Heinz’s definition, even though it does consist of the consecutive numbers from 1 to 2n.

Marián Trenkler, Obzory Matematiky, Fyziky a Informatiky, 1998, no. 51, pp.1-7, Magic Stars

10-A10-A

1

2

3 4

56

7

8

9

10

1112

13

14

15

1617

18

19

20

1

2

3

4

5

6

7

8

9

10

1112

13

14 15

16

17

18

19

20

10SM MT10


Magic star.. 3-D

Magic squares have a 3-D version, the magic cube. Is it possible for magic stars to also come in 3 dimensions? The answer is yes. The simplest version is an 8 pointed star consisting of two interconnected regular tetrahedrons.

There is one solution, using the numbers from 1 to 17, but without the 3, 9 and 15.It has 12 lines of 3 numbers all summing to 27.

This is the only solution possible (except for it’s complement) using this number set (not counting rotations or reflections). The magic constant, 27, is the smallest possible for a magic star of this type.

The three unused numbers 3, 9 and 15 may be incorporated in the pattern as follows:

Place the number 9 in the center of the star. It forms a magic line with it's two 'satellites', the 3 and the 15. The 9 also forms magic lines in conjunction with each of the 4 star point pairs, and with each of the 3 star midline pairs.

Thus we have a pattern using the consecutive numbers from 1 to 17, forming 22 lines of 3 numbers, all summing to 27

The following diagram is a little confusing. Try to visualize a tetrahedron with the apex, 14, pointing away, and another tetrahedron with the apex, 4, pointing towards you. The 3, 9 and 15 are not part of the magic star, but do enhance the pattern.

Aale de Winkel helped to visualize the pattern for an 8-point 3-D star. He also came up with this solution when Heinz indicated that numbers 1 to 15 and 1 to 16 didn’t work

After I (Heinz) determined that it was impossible to construct a 10-point 3_D star, Hermann Mierendorff provided an impossibility proof.

See the 3-D page on the Heinz Web site for more information, better diagrams, and two photographs of a wood block model.

H.D.Heinz, http://www.geocities.com/~harveyh/3-d_star.htm.htm

….. Magic star..3-D … 89

….. Magic star.. 3-D

87 - The 3-D order-3 magic star.

Magic Sum

The value each row, column, etc., sums to is called the magic sum. It is denoted by S. For a normal n-dimensional magic hypercube of order-m the sum is m (mn + 1).

See constant and Summations.

For a magic star, S is the sum of the numbers in each line. For a normal magic star of order-m, S = 4n + 2.

15

3

4

8

14

2

12

17

7

13

6

1

16

105

11

9


….. Magic Sum

Correct Summations Required

Magic Square

Regular

Magic Cube

Regular

Magic Tesseract

Regular

m rows m2 rows m3 rows

m columns m2 columns m3 columns

2 diagonals m2 pillars m3 pillars

4 3-agonals m3 files

8 4-agonals

Perfect Perfect Perfect

m rows m2 rows m3 rows

m columns m2 columns m3 columns

2m diagonals m2 pillars m3 pillars

4m2 3-agonals m3 files

6m2 2-agonals 8m3 4-agonals

12m3 3-agonals

16m3 2-agonals

88 - Comparison of magic squares cubes and tesseracts. Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published,1999, 0-9684700-4-1

Magic tesseract … 91

Magic tesseract

A magic tesseract is a four-dimensional array, equivalent to the magic cube and magic square of lower dimensions, containing the numbers 1, 2, 3, …, m4 arranged in such a way that the sum of the numbers in each of the m3 rows, m3 columns, m3 pillars, m3 files and in the eight major quadragonals passing through the center and joining opposite corners is a constant sum S, called the magic sum, which is given by: S = ½ m(m4+1) and where m is called the order of the tesseract.

A magic tesseract is also called a 4 dimensional hypercube.

A magic tesseract contains 54 number squares and 8 border number cubes, not necessarily magic.

89 - An order-3 magic tesseract shown in Hendricks projection with corners and center values only.

67

27 68

30

48

59 46

17

36

65 34

23

14

52 15

55

41


….. Magic tesseract

The full data for the above magic tesseract

65 24 34 31 71 21 27 28 68

22 35 66 72 19 32 29 69 25

36 64 23 20 33 70 67 26 30

(3, x, y, 1) (3, x, y, 2) (3, x, y, 3)

6 43 74 80 3 40 37 77 9

44 75 4 1 41 81 78 7 38

73 5 45 42 79 2 8 39 76

(2, x, y, 1) (2, x, y, 2) (2, x, y, 3)

52 56 15 12 49 62 59 18 46

57 13 53 50 63 10 16 47 60

14 54 55 61 11 51 48 58 17

(1, x, y, 1) (1, x, y, 2) (1, x, y, 3)

90 - The above order-3 magic tesseract in tabular form.

The center cell contains the number 41 (on all order-3 normal magic tesseracts). The 16 corner values are shown in bold type. One of the 8 quadragonals is 27 + 41 + 55.

There are 58 basic magic tesseracts of order-3. Each of these has 384 variations (aspects) due to rotations and reflections.

C.Planck (W.S.Andrews, Magic Squares and Cubes,1917, pp

363-375) refers to these as octahedroids, and their space diagonals as hyperdiagonals.

….. Magic tesseract … 93

….. Magic tesseract

Comparing Order-3 Hypercube Dimension Facts

Dimension Correct lines

# of Basic Aspects

2 8 1 8

3 31 4 48

4 116 58 384

5 421 2992 3840

6 1490 543328 46080

91 - Order-3 hypercube comparison. J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999,

0-9684700-0-9

J. R. Hendricks, All Third Order Magic Tesseracts, self-published 1999, 0-9684700-2-5

1

2

34

5

6

7

8

9 10

11

12

13

14

15

16

17

18


Magic triangular regions

Let Tn be the nth triangular number. Arrange the numbers from 1 to Tn in an equilateral triangular array. If all the triangular regions of 3 rows within this array sum to the same value, it is called a magical triangular region (MTR).

Fig. A shows an order-4 MTR using numbers from 1 to Tn (Tn = 10, n is called the order). There are Tn-2 equilateral triangles within this figure that sum to the same value of 28. One (shown) has the apex 6, another with apex 8, and the other with apex 2. Fig. B is an order-6 with 10 such regions. Shown is the one with apex 21. Also shown is one of three inverted constant-sum regions making this a type-E MTR. Not shown is an order-5 MTR.

92 - Magic triangular regions, orders 4 and 6. Usiskin & Stephanides, J. Recreational Mathematics, 11:3, 1978-79, pp.176-179, Magic Triangular Regions of Orders 4 and 5.

Katagiri & Kobayashi, J. Recreational Mathematics, 15:3, 1982-83, pp200-208,

Magic Triangular Regions of Orders 5 and 6

Main Diagonals

The two diagonal series of cells that go from corner to corner of the magic square. Each must sum to the constant in order for the array to be magic. The leading (or left) diagonal is the one from upper left to lower right. The right diagonal is the one from lower left to upper right.

1

2

3

4

5

6

7

8

9 10

13

1

2

3

4

5

6

7

8

9

10

11

12

14

1516

17

18

19

20

21

A. B.

Mapping … 95

Mapping

Mapping is a transformation of one image into another one. See also Graphs, Isomorphic magic stars and Magic circles.

93 - Transforming a 2-d magic star to 3-D cube and octahedron.

The cube is face perimeter magic with the vertices mapping to the triangles of the star. The octahedron is vertex magic with the edges of the eight triangles mapping to the star triangles.

94 - An order-4 magic square mapped to a tetrahedron.

1

2

3

4

5

6 7

8

9

10

11

121

2

3

4

56

7

8

9

10

11

12

1

2

3

4

5

67

8

910

11

12

1

2

3

4 5

6

7

8

9

10

11

12

13

14

15

16

1

2

3

4

5

6

78

9

10

1112

13

14

15

16


Modular arithmetic

Any system of arithmetic in which any two numbers are equivalent when they differ by an integral multiple of the modulus. For example, one writes:

x ≡ y (mod m)

means that x-y is divisible by m. Sometimes, this is called clock arithmetic, which has a modulus 12. Because of the modulus, this is often called a congruence equation and the regular congruence symbol with three bars is used instead of the equal sign.

Collins Dictionary of Mathematics. Hendricks, et al, uses modular equations to find the digits of a number at a given location in a magic square, cube, or hypercube.

Modular space

Instead of having the customary infinite continuous mathematical space for studying magic squares, etc., a lattice, or framework is designed sufficient to hold the numbers in a square array, or a cubic array. The x-axis is then of finite length and is selected to be the same as the order of the magic square, or cube. This way, there is always a position for each number and a number at each position. However, the x-axis is bent around and joins itself in a circle. The order m is modulo m. The y-axis undergoes the same treatment. One ends up with the best representation of a 2-dimensional modular space is on the surface of a donut. The x-axis goes around the ring, torus, or donut one way and the y-axis perpendicular to it the other way. The advantage of modular space is that all diagonals become continuous. The disadvantage is that it seems strange at first.

See coordinates, n-agonals, and Triagonals.

Monagonal

The orthogonal lines of a magic rectilinear object. Also called i-rows, 1-agonals, summations, orthogonal.

Most-perfect magic square … 97

Most-perfect magic square

A normal pandiagonal magic square of doubly-even order with two added properties. Any two-by-two block of adjacent cells (including wrap-around) sum to the same value which is 2m2+2, where m is the order of the magic square, and the integers come in complementary pairs distanced ½m along the diagonals.

Most-perfect magic squares can be precisely enumerated because the have a one-to-one relationship with Reversible squares and the number of reversible squares may be easily calculated.

All order-4 pandiagonal magic squares are most-perfect, but for orders greater then 4 an increasingly smaller percentage of pandiagonal magic squares are most-perfect.

1 63 3 61 12 54 10 56

16 50 14 52 5 59 7 57

17 47 19 45 28 38 26 40

32 34 30 36 21 43 23 41

53 11 55 9 64 2 62 4

60 6 58 8 49 15 51 13

37 27 39 25 48 18 46 20

44 22 42 24 33 31 35 29

95 - An order-8 most-perfect magic square. K. Ollerenshaw and D. Brée, Most-Perfect Pandiagonal Magic Squares, IMA 1998, 0-905091-06-X

Ian Stewart, Mathematical Recreations, Scientific American, November 1999

Note that both these authors use the series from 0 to m2-1 for mathematical convenience. The sum of each 2 by 2 square array is then 2m2-2. They also use n to indicate the order. In keeping with the rest of this book, we use m for this purpose and reserve n (when required) for dimension.

H.D.Heinz, http://www.geocities.com/~harveyh/most-perfect.htm


Multiplication magic square

A magic square where the constant is obtained by multiplying the values in the cells.

The magic square shown here uses 4 unusual arithmetic series. When the four cells in each row, column or diagonal are multiplied together, a magic product of 401,393,664 is obtained. When the order of the digits in each number is reversed, a new multiplication magic square is formed with a product of 4,723,906,824.

408 336 244 12 804 633 442 21

122 24 306 448 221 42 603 844

36 488 112 204 63 884 211 402

224 102 48 366 422 201 84 663

A. B.

12 24 36 48 21 42 63 84

102 204 306 408 201 402 603 804

112 224 336 448 211 422 633 844

122 244 366 488 221 442 663 884

The four arithmetic series for each of the above two squares.

96 - An order-4 multiplication magic square (not geometric) and its reverse.

This remarkable square was constructed by R. B. Edwards, an amateur magician of Rochester, New York. It is the nucleus of an order-6 Bordered adding magic square. See Order, doubly-even.

See geometric magic square for information on the two most common types of multiplication magic squares. See also Division magic square.

Joseph S. Madachy, Mathematics On Vacation, Nelson, 1966, 17-147099-0,

pp.89-90.

N (or n)… 99

N N (or n)

n is used for many things in mathematics. In Geometry it is often used for the dimension. In magic squares it has been used for the order. And, when you are counting it may stand for the number of objects. When it comes to magic cubes, tesseracts and hypercubes n could very well stand for each of these. So for higher-dimensional spaces Hendricks (and this book) uses n for the dimension, m for the order and N for the number of things. Note, however, that this book still uses n to indicate the order of magic stars.

n-agonals

A line going from 1 corner, through the center to the opposite corner, of a magic hypercube greater then dimension-2. Also called space diagonal. (A 1-agonal is an i-row (orthogonal line).)

Number of broken n-agonals for each continuous one

n 2 seg. 3 segments 4 segments Total

2 M–1 0 0 m

3 3(m-1) (m-1)(m-2) 0 m2

4 2(5m-8) 2(2m2-7m+7)

(m-1)(m-2)(m-3)

m3

97 – Number of segments in n-agonals for dimensions 2, 3, 4.

For each continuous n-agonal in n-dimensional space, there are a number of broken n-agonals, depending upon the order of the hypercube. There are 2 continuous diagonals in a square, 4 continuous triagonals in a cube, and 8 continuous quadragonals in a tessseract. So, the numbers in the table must be multiplied by the number of continuous ones in order to determine how many and of which kind of n-agonals are in a hypercube. These numbers only apply in conventional mathematical space. If modular space is used, then all broken n-agonals become continuous.

See Modular space, Triagonal and Quadragonal.


n-Dimensional hypercube of order-m

An imaginary structured object in the format of a lattice containing mn cells and which resides in an n-dimensional modular space. An n-dimensional magic hypercube of order-m may have the same or lower order magic hypercubes within it.

n Hypercube

0 point

1 line segment

2 a square

3 a cube or hexahedron

4 a tesseract

greater then 4 a hypercube

98 - Hypercube has both a general and a specific meaning:

J. R. Hendricks, Inlaid Magic Squares and Cubes, Self-published 1999,

0-9684700-1-7

Nasic magic square

Now commonly called a pandiagonal Magic Square.

The term was first published by A. H. Frost in the Quarterly Journal of Mathematics, London, 1865 and 1878, pp 34 and 93.

W. S. Andrews, Magic Squares and Cubes, Dover, Publ., 1960, p365

Normal

When used in reference to a magic square, magic cube, magic star, etc, it indicates that the magic array uses consecutive positive integers starting with 1 and going to mn (or 0 to mn –1). An equally popular term for this condition is pure.

Normalized position

See Standard position.

Number of .. …101

Normalizing

Rotating and/or reflecting a magic square or magic star to achieve the standard position so the figure may be assigned an index number. This changes an aspect of the magic object to the basic orientation.

Number of ..

Anti-magic squares: see Anti-magic squares Aspects of a magic square: see Aspects. Aspects of a magic star: See Magic star, normal. Basic magic tesseracts: see Basic magic tesseract Correct lines in a tesseract: see Perfect tesseract Cubes in a 5-D hypercube; see Magic hypercube Diagonals in a tesseract: see Summations. Order-4 magic square types: see Dudeney groups Squares in a cube: see Magic hypercube Third order magic squares: see Enumeration-squares

Number square

A square containing different numbers but is not necessarily magic. One could extend this definition to higher dimensions.

1 2 3 2 1 9 3 6 1 1 1 9 3

4 5 6 4 3 8 5 2 9 9 2 5 7

7 8 9 6 5 7 7 8 4 3 8 5 1

A. B. C. D. 3 3 1 9

99 - Four different types of number squares

A. Numbers are in simple sequence, B. second and third rows are multiples of first row, C. the three rows are all 3-digit square numbers. D. an order-4 Perfect Prime Square. So named because all 4-digit numbers in the rows, columns and main diagonals are distinct 4-digit primes. There are thus 20 distinct 4-digit primes in this square. There are at least 24 other distinct 1, 2 and 3 digit primes also in this square.

See Perfect prime squares.

Carlos Rivera, http://www.primepuzzles.net/puzzles/puzz_004.htm


O Octants

The eight parts of a doubly-even order magic cube if you split the cube in half in each dimension. i.e. if you divide an order-8 cube in this fashion, the octants are the eight order-4 cubes positioned at each of the eight corners of the original cube.

J.R.Hendricks, Inlaid Magic Squares and Cubes, Self-published 1999, 0-9684700-1-7, pp 123-129

Odd order

The order is not divisible by 2, i.e. 3 (the smallest possible magic square), 5, 7, etc.

14 10 17 6 18

2 11 25 3 24

19 5 13 21 7

22 23 1 15 4

8 16 9 20 12

100 – This inlaid magic square consists of two odd orders.

This normal order-5 contains an inlaid order-3 magic square. J. R. Hendricks, The Magic Square Course, self-published 1991, p 32

Oddly-even order

See Singly-even.

Opposite cells … 103

Opposite cells

If the coordinates of a cell are (w,x,y,z) then the opposite cell would be located at (m+1-w, m+1-x, m+1-y, m+1-z) where m is the order. In general if xi is the coordinate, then xi is replaced by m+1-xi.

See Symmetrical cells.

Opposite corners

The two cells that are at the ends of an n-agonal are also at opposite corners of the hypercube.

A corner has coordinates which have either a “1”, or an “m” for each coordinate. Simply replace the one by the other to obtain the coordinates of the opposite corner. For example in a 5-dimensional magic hypercube, you have a corner at (1,m,1,1,m) so the opposite corner would be at (m,1,m,m,1).

See Coordinates, Magic tesseract and Perfect magic tesseract for examples.

Opposite short diagonal pairs

Two short diagonals that are parallel to but on opposite sides of a main diagonal and each containing the same number of cells. See Semi-pandiagonal.

J. L. Fults, Magic Squares, Open Court 1974, 0-87548-197-3

Order-3, Type 2 magic squares

A normal order-3 magic square consists of 3 series of numbers with both the horizontal step and the vertical step positive numbers. Because the numbers are consecutive, both steps equal one.

However, a magic square consisting of non-consecutive numbers may be constructed of three series where the vertical step is a negative number. Such a magic square is called a type 2 magic square. It is easy to identify because the number in the bottom left corner cell is smaller then that in the cell immediately above it.


….. Order-3, Type 2 magic squares

Harry J. Smith speculated on this type of magic square and coined the term in a letter to Dr. Michael Ecker dated Dec. 8, 1990.

1 2 3 2 7 6 1 3 5 3 7 8

4 5 6 9 5 1 4 6 8 11 6 1

7 8 9 4 3 8 7 9 11 4 5 9

A B C D

101 - Two order-3 sets of 3 series and the resulting magic squares.

A. This number square shows three series with both horizontal and vertical steps equal to 1.

B. The magic square using the three series in A. It is the only normal basic order-3 and is a type 1.

C. This number square shows three series with a horizontal step of 2 and a vertical step of –1.

D. The magic square using the three series in A. It is the smallest possible type 2 magic square. See Vertical step. Harry J. Smith’s web page at http://home.netcom.com/~hjsmith/

Heinz’s Type 2 page at http://www.geocities.com/~harveyh/type2.htm

Order-m

Hendricks (and this book) always uses m to indicate the order or number of cells per side of a magic square, cube tesseract, etc. n is reserved to indicate the number of dimensions of the magic object.

Order n

n traditionally indicates the number of cells per side of the magic square, cube, tesseract, etc. However, because he does so much work in multi-dimensions, Hendricks (and this book) uses m for this purpose.

For a magic star, n indicates the number of points, (and the order) in the star pattern.

Order, doubly-even … 105

Order, doubly-even

The order is evenly divisible by 4. i.e. 4, 8, 12, etc.

This is probably the easiest type of magic square to construct.

The following example shows an order-6 (singly-even) adding magic square, with an inlaid order-4 (doubly-even) multiply magic square. The order-6 magic sum is 1,355 and the order-4 magic product is 401,393,664. Neither magic square is normal. This square was devised by R. B. Edwards, an amateur magician of Rochester, New York.

See Doubly-even order for another example.

223 283 200 322 163 164

177 408 336 244 12 178

228 122 24 306 448 227

258 36 488 112 204 257

308 224 102 48 366 307

161 282 205 323 162 222

102 - A singly-even add magic square with an inlaid doubly-even multiply magic square. Joseph S. Madachy, Mathematics On Vacation, Nelson, 1966, 17-147099-0, p.89

Order, even

The order is evenly divisible by 2. Order 4 is the smallest even order magic square (it is also doubly-even).

See Even Order and Order, doubly-even for examples.

Order, odd

The order is not divisible by 2, i.e. 3 (the smallest possible magic square), 5, 7, etc.

See Odd order for an example.


Order, singly-even

The order is evenly divisible by 2 but not by 4. i.e. 6, 10, 14, etc. This order is by far the hardest to construct. See Order, doubly-even for an example.

Ornamental magic squares

A general term for magic squares containing unusual features. Some examples are; Bordered, Composition, Inlaid,

Overlapping, Reversible, Lozenge, Serrated, etc.

103 - A pentagram of five magic diamonds.

Each magic diamond sums to 162. In addition each diamond contains four 2 x 2 cells that sum to 162 as well as four 3 x 3 and one 4 x 4 diamonds with corner cells also summing to 162. Within each diamond, each number ends with one of two digits. Notice the digits 1 to 5 at the points. (Illustration adapted from W.S. Andrews).

W. S. Andrews, Magic Squares and Cubes, Dover, Publ., 1960, p.172

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50 31

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30

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69

52

32

912

29

49

72

79

42

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73

8

38

58

6343

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48

683

53

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37

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66

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7516

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45

76

Ornamental magic stars … 107

Ornamental magic stars

Any Magic Star containing unusual features. It may have one star embedded in another, more then four numbers to a line, consist of prime numbers (or any unusual number series), etc.

104 - This ornamental magic star consists of two interlocked order-8, pattern B magic stars.

The inner star is a normal magic star in standard position and is, in fact, index # 30. The outer star is also magic but is not normal because the numbers are not consecutive and the two inside cells of each line are in the interior of the star. It sums to 68 in each line. The normal star, of course, sums to 34.

See an order-9 ornamental star on the back cover of this book.

H.D.Heinz, http://www.geocities.com/~harveyh/unusualstr.htm.

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Orthogonal

Lines that are perpendicular to each other. In the magic square, the rows and columns are orthogonals.

105 - An order-3 cube showing coordinates and line paths

This cube shows the orthogonals, the rows, columns and pillars. Note that in each case only 1 coordinate changes as you move along the line. These lines are collectively also called i-rows, 1-agonals, or monagonals.

Shown for contrast is a diagonal (311, 322, 333) where 2 coordinates change as you move along it. Also shown is a triagonal (111, 222, 333), which passes through the center of the cube and requires a change of all 3 coordinates.

For brevity, no brackets or commas are shown for the coordinates.

Orthomagic square of squares.

See Square of squares

row (x)111

133

123

311211

113

112

313

222

113

pillar (z)

column (y)

diagonal

triagonal

333

322

Overlapping magic squares … 109

Overlapping magic squares

A special type of inlaid magic square where 1 square partially (or completely) overlaps another magic square (probably of a different order).

71 1 51 32 50 2 80 3 79

21 41 61 56 26 13 69 25 57

31 81 11 20 62 64 18 63 19

34 40 60 43 28 65 17 55 27

48 42 22 54 39 75 7 10 72

68 53 15 33 16 44 58 77 5

14 29 67 49 66 24 38 59 23

76 37 70 73 8 4 36 30 35

6 45 12 9 74 78 46 47 52

106 - This order-9 overlapping magic square by L. S. Frierson includes orders 3, 5, and 7. J.R. Hendricks, Frierson’s Fuddle (Problem #1945), Journal of Recreational Mathematics, 25:1, 1993, p77

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P Palindromic magic square

Palindromes are numbers (or letters) that read the same right-to-left as left-to-right. Palindromic magic squares may be any type of magic square, but consisting only of palindromes.

363 424 646 747 757 767 787 393

696 232 383 898 939 969 242 525

676 949 222 595 737 888 272 545

656 868 959 666 444 373 353 565

636 343 484 333 999 626 878 585

535 292 777 848 262 555 929 686

494 979 838 323 282 252 989 727

828 797 575 474 464 454 434 858

107 - An order-8 bordered palindromic magic square by A. W. Johnson, Jr.

The magic sums are also palindromes. S4 = 2442, S6 = 3663 and S8 = 4884.

A. W. Johnson, Jr., Journal of Recreational Mathematics, 21:2, 1989, p.155.

Pandiagonal

Pandiagonal means “all diagonal”, which means that the broken diagonals are also included. Sometimes pan-2-agonal is used, instead, especially in n-dimensional space. A 2-agonal is described through space if any two coordinates change while the rest remain constant.

For example in a cube of order 4, one could describe a diagonal through (1,2,3) by holding y constant while x and z are allowed to change. Such a set could be:

(1,2,3) ; (4,2,4) ; (3,2,1) ; and (2,2,2)

….. Pandiagonal … 111

….. Pandiagonal

In this example x is decreasing in increments of one and z is increasing by increments of one and all coordinates are kept within the modulus 4. There are N = n!.mn-1/(n-2)! diagonals in an n-dimensional magic hypercube of order m, including the broken ones.

The broken diagonals in a magic square consist of two elements. In a magic cube there are 2 or 3 element broken triagonals. In a magic tesseract they may are 2, 3, or 4 element broken quadragonals. Etc.

See Broken diagonal pair for an illustration

Pandiagonal magic cube

A Pandiagonal Magic Cube has the normal requirements of a magic cube plus the additional one that all the squares (planes) also be pandiagonal. Remember that an ordinary magic cube does not require even the main diagonals of these squares to be correct.

There are 9m2 + 4 lines that sum correctly (m2 rows, m2 columns, m2 pillars, 4 main triagonals and 6m2 Diagonals). Order-7 is the smallest possible order of pandiagonal magic cube. This is the original definition of a Perfect Magic Cube.


Pandiagonal magic square

Also known as Diabolic, Nasic, Continuous, Indian, Jaina or Perfect. To be pandiagonal, the broken diagonal pairs must also sum to the constant. This is considered the top class of magic squares. Some pandiagonal magic squares are also associative (order 5 & higher) . Because of the vast number of combinations possible, individual magic squares may contain unique features, which make them more magic.

There are 4m lines that sum correctly (m rows, m columns and 2m diagonals).

There is only 1 basic order 3 magic square and it is not pandiagonal.

Of the 880 basic order 4 magic squares, only 48 are pandiagonal and none of these are associative.


….. Pandiagonal magic square

Order 5 has 3600 basic pandiagonal magic squares (Only 36 essentially different).

Order 7 has 678,222,720 basic pandiagonal magic squares, of which 38,102,400 are regular and 640,120,320 are irregular.

Order 8 has more then 6,500,000,000 pandiagonal magic squares.

There are NO singly-even pandiagonal magic squares All the above numbers assume we are considering only normal, basic magic squares.

1 7 18 24 15 12 3 9 20 21

19 25 11 2 8 10 16 22 13 4

12 3 9 20 21 23 14 5 6 17

10 16 22 13 4 1 7 18 24 15

23 14 5 6 17 19 25 11 2 8

A. B.

108 - An essentially different pandiagonal magic square and one of its 100 transformations.

A. is the original pandiagonal magic square. B is a new pandiagonal magic square obtained by moving the two top rows to the bottom. A pandiagonal magic square is a Perfect magic

square. See Essentially different and Regular and irregular.

Pan-magic stars

An order-8A type magic star then can be constructed by a systematic transformation of odd-order pandiagonal magic squares greater then order-5. The outside diagonals of the magic star are formed from pandiagonal pairs one member of which is a corner cell.

Aale de Winkel investigated this type of magic star in the spring of 1999 which later resulted in his and my joint investigation of Iso-like magic stars. (Iso-like magic stars do not require that the generating square be pandiagonal, but instead uses plusmagic or diamagic patterns of a Quadrant magic square.)

….. Pan-magic stars …113

….. Pan-magic stars

Unlike iso-magic stars which cannot use all the numbers, pan-magic stars may use all the numbers in the originating magic square but require the use of duplicate numbers to complete the pattern. In the following example, I use shading to indicate the duplicate numbers.

A variation of pan-magic stars is what de Winkel calls the butterfly. See Iso-like magic stars for a comparison of Iso-like, Isomorphic and Pan magic star features.

109 - An order-7 pan-magic star.

The order is determined by the generating magic square (see next page) and is the number of cells per line.

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….. Pan-magic stars

1 19 30 48 10 28 39

49 11 22 40 2 20 31

41 3 21 32 43 12 23

33 44 13 24 42 4 15

25 36 5 16 34 45 14

17 35 46 8 26 37 6

9 27 38 7 18 29 47

110 - The order-7 pandiagonal magic square used for this pan-magic star. H.D.Heinz, http://www.geocities.com/~harveyh/panmagic.htm

See more on this subject at Aale de Winkels Web site at http://www.adworks.myweb.nl/Magic/

Pan-n-agonal.

All coordinates are changing in unit intervals either plus, or minus, as one describes a path through space. In an n-dimensional magic hypercube of order m, the number of r-agonals, where 1<r<n+1 is given by N which is:

N = 2r-1.nCr.mn-1

where m is the order, n is the dimension, r is the space diagonal of the rth dimension and where C stands for the customary combinations.

Panquadragonal

Broken quadragonal pairs that are parallel to a quadragonal and that sum to the magic constant. If all these pairs sum correctly, the magic tesseract is panquadragonal. It is analogous to a pandiagonal magic square but instead of moving a row or column from one side to the other and maintaining the magic properties, you move any cube from one side to the other. When one moves along the panquadragonal, 1 cell at a time, four coordinates change. See also, Pantriagonal.

J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1999 J.R.Hendricks, The American Mathematical Monthly, Vol. 75, No. 4, April 1968,

p.384

Pantriagonal … 115

Pantriagonal

Sometimes called Pan-3-agonal.

This term is used for cubes, or high dimensional hypercubes. In n-dimensional space, if any three coordinates are changing while the rest remain constant, then one describes a triagonal through space, of which most are broken. The main triagonal is the one which passes through (1,1,1) and has successive coordinates (2,2,2),…, (m,m,m) in a cube.

In N-dimensional space, the n-agonal may be broken into as many as n segments. For magic cubes there are:

• 4 continuous triagonals

• 12(m-1) triagonals broken into pairs, and

• 4(m-2)(m-1) triagonals broken into 3 sections.

If all the broken Triagonal lines sum correctly, the magic cube is pantriagonal.

See Orthogonal and Pan-triagonal magic cube for illustrations. See n-agonals and Triagonals for tables.


Pan-triagonal magic cube

If all triagonal pairs (Pan-triagonals) sum correctly, the magic

cube is pantriagonal.It is analogous to a pandiagonal magic square but instead of moving a row or column from one side to the other and maintaining the magic properties, you may move any plane from one side to the other.

There are 7m2 lines that sum correctly (m2 rows, m2 columns, m2 pillars, and 4m2 triagonals). There may be some diagonals in the cube but they are not required. Order-4 is the smallest possible order pantriagonal magic cube. See also, Pandiagonal magic

cube.

There are 6 fundamental pandiagonal magic squares of order-4 from which the 48 basic squares are derived. There are 20 fundamental pantriagonal magic cubes of order-4 from which the 160 basic cubes are derived.

J.R. Hendricks, Pan-3-agonal Magic Cubes of Order-4, JRM, 13(4), 1980-81, pp

274-281


….. Pan-triagonal magic cube

111 - A pan-3-agonal magic cube of order 4.

One of the four triagonals is 60 + 46 + 5 + 19= 130.

One of the 15 pan-triagonals parallel to it is 6, 17, 59, 48 and is broken into two parts. Another one of the 15 which is broken into three parts is 36, 10, 29 55.

See Broken diagonal pairs and Pantriagonals.


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Parity patterns … 117

Parity patterns.

Parity patterns are the arrangements of the odd and even numbers in a magic square. Some form pleasing patterns, many do not. The most pleasing patterns are usually symmetrical.

See Lozenge and Self-similar for other patterns.

6 3 19 25 12

20 22 11 8 4

13 9 5 17 21

2 16 23 14 10

24 15 7 1 18

112 - Order-5 pandiagonal magic square with odd numbers underlined.

Partitioning

Sub-dividing. A square may be sub-divided into cells, as shown below.

113 - An order-4 square partitioned into cells.

Similarly, a cube may be sub-divided into building blocks, There are m2 cells in a square of order m and m3 cells in a cube of order m. There are 3m squares in the cube which may be found by this partitioning.

In four-dimensional space, the equivalent partitioning yields, m4 cells, 4m cubes and 6m2 squares,


Patchwork magic squares

An Inlaid magic square that has magic squares or odd magic shapes within it. The most common shape is a magic rectangle, but diamonds, crosses, tees and L shapes are also possible. These shapes are magic if the constant in each direction is proportional to the number of cells. For example, a 4 x 6 rectangle may have the constant of 100 in the short direction and 150 in the long direction. Diagonals are not required to be magical (except for squares).

This example by David Collison is an order 14 magic square, containing 4 order 4 magic squares in the quadrants, a magic cross in the center, 4 magic tees on the sides, and 4 magic elbows in the corners.

The order-14 magic sum is 1379 and dividing by 14 gives the mean for each cell as 98.5. Adding the numbers in each line of a generalized part and dividing by the number of cells in the line will, in each case, give the value of 98.5!

154 155 41 44 2 6 190 192 8 193 38 35 161 160

42 43 156 153 195 191 5 7 189 4 159 162 37 36

40 157 91 105 104 94 3 194 83 113 112 86 163 34

158 39 102 96 97 99 196 1 110 88 89 107 33 164

177 20 98 100 101 95 140 57 90 108 109 87 171 26

24 173 103 93 92 106 58 139 111 85 84 114 167 30

176 23 178 17 137 136 59 131 65 63 172 27 29 166

174 21 19 180 60 61 66 138 132 134 25 170 31 168

22 175 75 121 120 78 135 62 67 129 128 70 165 32

18 179 118 80 81 115 133 64 126 72 73 123 28 169

146 51 82 116 117 79 188 9 74 124 125 71 45 152

52 145 119 77 76 122 11 186 127 69 68 130 151 46

54 55 144 141 187 183 13 15 181 12 147 150 49 48

142 143 53 56 10 14 182 184 16 185 50 47 149 148

114 - Collison’s order-14 patchwork magic square contains magic squares, elbows, tees and a cross.

J. R. Hendricks, The Magic Square Course, self-published 1991, p 312.

Pathfinder … 119

Pathfinder

An orderly and systematic way to find one’s way through n-dimensional space. Through any given element, or cell, there are (3n-1)/2 different paths., or lines, For a square, this means that there are 4 paths which are a row, a column and two (broken, if needed) diagonal ways. Through a cell of a cube, there are 13 routes. Through a tesseract, there are 40. One may travel forwards, or backwards on any route, or path. The method is found in Magic Squares to Tesseracts by Computer.

See Coordinate iteration.

J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999, 0-9684700-0-9

Pattern

There are many types of patterns involved in magic squares, cubes, etc. In fact, the square, cube, star, etc is itself a pattern. See Algebraic, Complimentary pair, Dudeney Groups, Magic

lines and Parity patterns.

Perfect magic cube

A perfect magic cube is pantriagonal and all of its planes (the magic squares) are pandiagonal. There are 13 m2 lines that sum correctly (m2 rows, m2 columns, m2 pillars, 4m2 triagonals and 6m2 diagonals). Order-8 is the smallest possible order perfect magic cube.

J.R.Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999. 0-9684700-0-9

*** This is a new definition. ***

An older definition of a Perfect Magic Cube is what Hendricks now calls a Pandiagonal magic cube. This older definition probably originated from the fact that a pandiagonal magic square is perfect. However, perfect is now construed to mean that it is pandiagonal but all lower order magic objects within it are perfect. This makes the definition consistent for all dimensions.

For an Order-8 pandiagonal first published in 1888, see Benson & Jacoby, Magic

Cubes New Recreations, Dover, 1981,0-486-24140-8.


Perfect magic square

Another (but not commonly used) name for Pandiagonal magic square. However, this name shows the relationship of the highest class of rectilinear magic figures, the perfect square, perfect cube, perfect tesseract, etc. See Magic sum for a comparison table.

26 3 21 33 45 15 32

38 48 28 36 4 9 12

30 39 2 13 49 22 20

19 14 43 25 37 6 31

10 23 41 7 8 46 40

34 1 11 44 27 42 16

18 47 29 17 5 35 24

115 – Order-7 with an inlaid order-5 pandiagonal magic square. J. R. Hendricks, The Magic Square Course, self-published 1991, p 185

Perfect magic tesseract

John R. Hendricks constructed the first perfect magic tesseract (order-16) in 1998. It was confirmed correct by Clifford Pickover in 1999. He later published the equations for a 5-dimensional perfect magic hypercube of order-32. However, as it contains the numbers 1 to 33,554,432, he thought it impractical to publish the hypercube itself!!

A tesseract is a 4-dimensional hypercube. It is perfect if all pan-quadragonals are correct, and all the magic squares and magic cubes within it are perfect. i.e. the magic squares are all pandiagonal and the magic cubes are all pantriagonal and pandiagonal. There are 40m3 lines that sum correctly. They are m3 rows, m3 columns, m3 pillars, m3 files, 8m3 quadragonals, 16m3 triagonals, and 12m3 diagonals.

The smallest order perfect tesseract is order-16 and is too big to reproduce here. However, it is practical to show the outline diagram with the corner numbers.

….. Perfect magic tesseract … 121

….. Perfect magic tesseract

116 - Values of the corner cells of the order-16 perfect tesseract.

This tesseract utilizes the numbers 1 to 65536 and contains:

49,152 diagonals 65,536 triagonals 32,768 quadragonals 16,384 rows, columns, pillars and files 163,840 ways to sum to 524,296

It also contains 64 perfect magic cubes, and 1536 perfect magic squares, all of order-16.

21383

16842

7317

30584

25913

37917

34388

56079

51522

65536

60849

45294

41635

3804

14438

10795


….. Perfect magic tesseract

With any order perfect magic hypercube, any element can serve as the starting point on an axis reference system. There are mn elements and therefore mn

different hypercubes for each essentially different hypercube. As each of these have 2nn! aspects, one essentially different perfect hypercube will generate m

n2nn! apparently different hypercubes.

*** Perfect magic tesseract is a new definition. *** Please review the revised definition for the Perfect magic cube. These new definitions are more compatible with that of a perfect (pandiagonal) magic square.

J.R.Hendricks, Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-

published,1999, 0-9684700-4-1 J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999,

0-9684700-0-9

Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton University Press, 2002, 0-691-07041-5, page 121.

Heinz-Private correspondence with Hendricks and Pickover, 1999.

Perfect prime squares.

A perfect prime square is not a magic square but a number square where all rows and columns and main diagonals consist of distinct prime numbers, reading in both directions, with a length equal to the order of the square. Of course, with an array so full of the odd digits, the square will also be rich in smaller primes.

Originally discovered by L.E. Card in 1968 and rediscovered in 1998 by Carlos Rivera and popularized on his Primes Puzzles page.

The order-11 (next page) was found in late 1998 by T. W. A. Baumann of Germany along with many of smaller orders seven to ten. It contains 48 different 11-digit prime numbers and appears on Rivera’s PP&P Web page. There are no order-3 perfect prime squares because all contain duplicate 3-digit primes, but such squares exist for all other orders from 2 to at least 11.

….. Perfect prime squares … 123

….. Perfect prime squares.

3 7 9 7 9 9 1 3 9 7 3

7 9 1 9 1 9 1 7 9 9 9

7 1 1 9 1 9 3 9 7 9 9

1 1 1 1 3 7 9 9 7 7 1

1 1 1 7 1 7 1 9 3 3 1

1 7 3 7 1 7 9 3 7 1 1

1 1 7 3 1 1 7 9 9 1 3 1 1 3 3 3

7 2 2 5 1 3 9 1 9 1 9 1 1 3 3 7

3 1 8 5 9 7 7 9 9 7 1 1 3 7 9 1

9 0 7 9 3 7 9 3 3 3 7 7 7 7 3 9

3 9 1 1 3 3 3 9 3 3 9 1 3 9 1 3

L. E. Card T. W. Baumann (Rivera)

117 - Order-5 and order-11 perfect prime squares L. E. Card, J. Recreational Mathematics, 1:2, 1968, pp.93-99. Carlos Rivera, http://www.primepuzzles.net/puzzles/

Perimeter magic polygons

A PMP is defined as a regular polygon with consecutive numbers from 1 to n placed along the perimeter in such a way that the sums of the integers on each side (edge) is a constant. The order refers to the numbers along each side of the figure.

There are 18 basic order-4 perimeter magic triangles. Their magic sums are; 17, 19, 20, 21 and 23. There are 4 basic order-3 perimeter-magic triangles with sums of 9, 10, 11, and 12.

The examples in figures 118 and 119 are edge perimeter magic. See Mapping for face perimeter and vertex perimeter magic examples.


….. Perimeter magic polygons

118 - Two of 18 order-4 perimeter-magic triangles.

Fig. B, constructed by D.M. Collison, is bimagic. If each number is squared, the triangle is still perimeter magic. Perimeter magic can also be applied to higher dimensions. See Mapping for examples of a magic tetrahedron and octahedron.

J. R. Hendricks, The Magic Square Course, self-published 1991, pp.1-4

119 - An order-5 pentagon and an order-3 septagon. Terrel Trotter, Jr., Journal of Recreational Mathematics, 7:1, 1974, pp.14-20,

Perimeter-magic Polygons.

1

2

3

4

5 6

7

8

9

10

11

12

13

14

1

3 4

5

6

7

8

910

11

12

14

1516

17

18

19

20

13

2

45 22

1

2 3

4

5

6

7 8

91

2

3 4

5

6

7

8

9

A. B.

Perimeter anti-magic octahedrons … 125

Perimeter anti-magic octahedrons

As all the faces of a polyhedron are triangles, it is impossible to place distinct integers on its vertices so that every triangle will have the same perimeter sum. Therefore, no such polyhedron can be magic. However, if the perimeter sums are all different the polyhedron will be perimeter anti-magic.

In the case of an octahedron it turns out that there are 15 basic ways to distribute the digits 1 to 6 on the vertices with eight different perimeter sums. Of these 15, five produce distinct perimeter sums for the eight triangles and so are perimeter anti-magic.

120 - Three of the 5 anti-magic Octahedrons.

Figures A. and B. may each be complemented by subtracting each number from 7, to provide another solution. C. is self-complementing.

C. W. Trigg, Journal of Recreational Mathematics, 11:2, 1978-79, pp.105-107, Perimeter Anti-magic tetrahedrons and Octahedrons.

Pillars

The Z dimension in a coordinate system of addressing the cells in a magic cube or higher order hypercube. (x = rows and y = columns.) See orthogonals.


A. B. C.

2

3

4

56

1

23

45

6

1 1

23

4

5

6


Plusmagic

An pattern of n cells in the shape of a plus sign that appears in each quadrant of an order-n quadrant magic square.

121 - Plusmagic Quadrant pattern for order-9 and order-13 H. D. Heinz, http://www.geocities.com/~harveyh/quadrant.htm

Prime number magic squares

A magic square consisting only of prime numbers. They are not too difficult to construct. The difficulty is in constructing ones consisting of consecutive primes. The first order 3 magic squares of this type was only published in 1988 and consists of nine, 10 digit primes. The author proved there are only two such squares with prime numbers under 231.

Harry L. Nelson, J. Recreational Mathematics, 20:3, 1988, p.214-216.

In 1913 it was proved (?) (Scientific American vol.210, no.3 pp. 126-7) that it is impossible to construct a consecutive prime number magic square of order smaller then 12. The order 12 magic square shown by the author, however, contained the digit 1 and missed out the digit 2. (Of course the number 1 is no longer considered a prime, and the number 2 can never appear in a prime number magic square, because it is the only even prime, and parity would be destroyed.)

All prime magic figures have a complement solution (see complement magic squares). However, when you complement a prime number magic square, most resulting numbers will be composite, so the complement of a prime number magic square is almost certainly not a prime number magic square.

….. Prime number magic squares … 127

….. Prime number magic squares

Order of Consecutive Prime square

Lowest prime number in series

3 1,480,028,129

4 31

5 269

6 67

122 – Starting prime # for consecutive primes magic square.

2621 2477 2039 1289 3251 1583 3533 2207

3257 1361 3491 2393 2333 2963 1709 1493

2609 1811 2837 2087 2687 1889 2939 2141

2777 2819 2753 1823 1223 3701 1931 1973

2351 2879 1049 3527 2927 1997 1871 2399

1283 2339 2861 2063 2663 1913 2411 3467

1559 3041 1259 2357 2417 1787 3389 3191

2543 2273 2711 3461 1499 3167 1217 2129

123 - A. W. Johnson, Jr.’s bordered, order-8 prime number magic square.

This order-8 is a simple prime magic square summing to 19,000. The order-6 is pandiagonal with a magic sum of 14,250; and the order-4 is symmetrical with a magic sum of 9,500.

See Vertical step for the smallest possible consecutive prime order-3 magic square.

A. W. Johnson, Jr., J. Recreational Mathematics 15:2, 1982-83, p. 84


Prime number magic stars

A magic star that consists only of prime numbers is (naturally) a prime number magic star. Because prime numbers are not consecutive natural numbers, these stars are not normal. However, they can consist of consecutive prime numbers, so there are two types of series to look for, minimal solution prime stars and consecutive primes, prime stars. And just as with normal magic stars, rotations and reflections do not count as unique.

124 - Order-5 prime magic stars. A. is minimal solution, B. is minimal consecutive primes solution.

The minimal solution for order-6 prime stars uses 12 of the 14 primes (not 37 and 43) from 3 to 47 with 8 basic solutions and a magic sum of 82. The minimal consecutive primes solution uses primes 29 to 73 with 20 basic solutions and a magic sum of 204.

An interesting point. Unlike all normal magic stars and all solutions for the order-5 prime magic star, none of the order-6 minimal solutions consist of complement pairs, and only 4 of the 20 minimal consecutive solutions have complement partners.

H. D. Heinz, http://www.geocities.com/~harveyh/primestars.htm

Principal reversible square

See Reversible square, principal

#6 of 12

7 2311 43

19

5

17

31

13

41#1 of 12

13907

13913

13967

14009

13933

13963

13999

13921 13997 13931

A. B.

Pure magic square … 129

Proper magic cube

Refers to a cube that contains exactly the minimum requirements for that class of cube. i.e. a proper simple or pantriagonal magic cube would contain no magic squares, a proper diagonal magic cube would contain exactly 3m + 6 simple magic squares, etc. See the table in Summations for minimum requirements. This term was coined by Mitsutoshi Nakamura in an email of April 15, 2004.

Pure magic star

See Magic square, normal.

Pure magic star

See Magic star, normal.

Pythagorean magic squares

It is possible to build a set of three magic squares on the sides of a triangle with sides equal to a pythagorean triplet. R. V. Heath published several such magic squares in 1933. Here are examples (Heinz) of two versions of this type of magic square built around the 3, 4, 5 pythagorean triangle.

The first set consists of three order-4 magic squares. The square of the magic sum of “a” plus the magic sum of “b” is equal to the square of the magic sum of “c”. The square of any cell in “a” plus the square of the corresponding cell in “b” equals the square of the same cell in “c”.

In addition, the square of the sums of “a” added to the squares of the sums for “b” for any 2 x 2 set of cells is equal to the square of the sum of the corresponding 2 x 2 set of cells in magic square “c”. The four corners of any 3 x 3 or 4 x 4 square (wrap-around works) also has this same property, as does the sum of all the cells in the squares.

The second example shows 3 simple magic square of orders 3, 4 and 5 built on the sides of the 3, 4, 5 triangle. The magic constants of “a” and “b” are 27 and 38 which sums to the magic constant of “c”.

Any addition magic square, when multiplied by members of a pythagorean triplet, will produce a pythagorean set of magic squares.

R. V. Heath, Mathemagic, Dover, 1953


….. Pythagorean magic squares

125 - Pythagorean magic squares, all order-4, Sc2 = sa

2 + Sb

2.

126 - Pythagorean magic squares, orders 3, 4, 5; Sc = sa + Sb.

9 2 25 18 11

3 21 19 12 10

22 20 13 6 4

16 14 7 5 23

15 8 1 24 17

6

10

13

5

8

7

1211

92

9

14

13

15

12

3

8

5

6

17

10

16

11

4

7

a b

c

5

3

24

39

36

42

33

6

21

1215

48

27

45

30

9

18 4

32

52

48

56

44

8

28

16

20

64

36

60

40

12

24

40 65 60

70 55 10 35

20 25 80 45

75 50 15 30

a b

c

Quadragonal … 131

QQQQ Quadragonal

A 4-dimensional version of the 2-dimensional diagonal and the 3-dimensional triagonal. However, just as a 2-dimensional diagonal can exist in spaces higher then 2 dimensions, and a triagonal in spaces higher then 3 dimensions, so also a quadragonal can exist in spaces higher then 4 dimensions.

A Magic Tesseract (4- dimensions) requires eight of these lines of n numbers summing correctly that go from one corner to the opposite corner through the center of the tesseract. Also called a 4-agonal.

127 - The eight opposite corner pairs of a magic tesseract. J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999,

0-9684700-0-9, p 116

1

1

2

2

3

3

4

4

5

56

6

7

78

8


Quadrant

A quarter of a magic square. The four quadrants are; upper-left, upper-right, lower-left and lower-right. If the magic square is even, the size of each quadrant is n/2 square. If the magic square is odd, the size of each quadrant is (m+1)/2 square and the center row or center column is common to two orthogonally adjacent quadrants. Quadrants figure prominently in Quadrant magic squares.

Quadrant magic pattern

An pattern of m cells that appears in each quadrant of an order-m quadrant magic square and is symmetrical around the center cell of the quadrant. There are 5 such quadrant patterns for order-5, 7 for order-9, 38 for order-13, and 253 for order-17.

The first patterns discovered were named. Later ones were identified with an index number. While patterns such as these can be readily found in magic squares, to qualify as quadrant magic, the square must contain an identical pattern in all four quadrants.

Many quadrant patterns have cells that are in common with the orthogonal adjacent arrays. This is because the center row of the magic square is common to the two top and bottom arrays.. Likewise, with the center column, which is common to the two side-by-side arrays.

128 - Some order-13 quadrant magic patterns. H. D. Heinz, http://www.geocities.com/~harveyh/quadrant.htm

Quadrant magic pattern – even order … 133

Quadrant magic pattern – even order

Even order quadrant magic squares have not yet been investigated. However, it is now known that they do exist. Any orders 8m magic square may be quadrant magic. However, two characteristics are markedly different.

The center cell of the quadrant is not part of the pattern. The pattern is still required to be symmetrical around the center point of the quadrant.

Cells are not common to two orthogonally adjacent quadrants. This is because there is no center row and center column of the magic square common to two quadrants as it is with odd order magic squares.

Of the original five first discovered (and named) quadrant patterns, only the lringmagic, sringmagic (no sringmagic for order-8) and crosmagic exist in even order. I (Heinz) estimate that when the subject is investigated, far fewer patterns will be found to exist for even order then for odd order.

XXXX X X X X XX X X XX XXXX X X

129 - Even order crosmagic and lringmagic patterns for order 8, sringmagic and lringmagic patterns for order 12.


Quadrant magic square

Some magic squares of orders m equal to 4m + 1, have patterns of m cells appearing in each quadrant that sum to the magic constant.

See Quadrant magic pattern.

If a magic square contains 4 of these patterns in the 4 quadrants, and if they are all the same type, it is called a quadrant-magic square.

Because the center row and the center column of the square is common to two adjacent quadrants, it is common for a border cell to be a member of to two different patterns. We have deliberately chosen the arrays to show below to avoid this.

Quadrant magic squares of odd order were investigated by Aale deWinkel and Harvey Heinz in 1999. It is now known that there are equivalent magic squares of even order. See Quadrant magic pattern- even order.

43 22 157 136 115 81 60 39 5 153 119 98 77

106 85 64 30 9 144 123 102 68 47 26 161 140

13 148 127 93 72 51 17 165 131 110 89 55 34

76 42 21 169 135 114 80 59 38 4 152 118 97

139 105 84 63 29 8 156 122 101 67 46 25 160

33 12 147 126 92 71 50 16 164 143 109 88 54

96 75 41 20 168 134 113 79 58 37 3 151 130

159 138 117 83 62 28 7 155 121 100 66 45 24

53 32 11 146 125 104 70 49 15 163 142 108 87

129 95 74 40 19 167 133 112 91 57 36 2 150

23 158 137 116 82 61 27 6 154 120 99 78 44

86 65 31 10 145 124 103 69 48 14 162 141 107

149 128 94 73 52 18 166 132 111 90 56 35 1

130 - This order-13 quadrant magic square is 14 times quadrant magic. i.e 14 patterns appear in all 4 quadrants.

….. Quadrant magic square … 135

….. Quadrant magic square

That is, there are 14 patterns that each appear in all four quadrants of the magic square. In fact, of the 38 total magic patterns possible for order-13, 32 of them appear at least once in the square and all 32 appear in the top left quadrant.

W

W X X X

W X

W W W W W W W X X X X X

W X

W X X X

W Z Z Z

Y Y Z Z

Y Y Y Y

Y Z Z Z

Y Y Y Y

Y Y Z Z

Z Z Z

131 - Some patterns of the order-13 quadrant magic square

The above diagram shows the above quadrant magic square in more detail.

We have indicated the center cell of each quadrant and the outline of the two smaller patterns. However, each pattern must be considered as occupying a 7 x 7 array so cells on the center column and center row belong to two adjacent quadrants. Also, each pattern appears in all four quadrants.

H. D. Heinz, http://www.geocities.com/~harveyh/quadrant.htm See more on this subject at Aale de Winkels web site at

http://www.adworks.myweb.nl/Magic/


RRRR Radix

See Base.

Regular magic square

See Associative magic squares

Also a major classification of Pandiagonal magic squares See Regular & Irregular

Regular & Irregular

A common method of constructing a Pandiagonal magic square makes use of 2 subsidiary squares where letters are used to represent various constants. The values in the two squares are then combined to obtain the value for the corresponding cell of the magic square. If each letter appears exactly once in each row, column and diagonal in both squares, the resulting pandiagonal is considered regular. If they do not appear an equal number of times in the rows, columns and diagonals of one or both squares, then the resulting pandiagonal is irregular. All pandiagonal magic squares of orders 4 and 5 are regular. There are 38,102,400 regular pandiagonals of order 7 and 640,120,320 irregular.

The number of cyclical pandiagonal magic squares of the mth order (m = prime) is (m-3)(m-4)(m!)2/8.

The term regular sometimes is used for associated magic squares. [1]

This type of pandiagonal magic square is sometimes called cyclical. [2]

[1] Andrews, W. S., Magic Squares & Cubes, 2nd edition, Dover Publ. 1960 [2] Benson & Jacoby, New Recreations with Magic Squares, Dover, 1976 0-486-23236-0, p93- 141)

….. Regular & Irregular … 137

….. Regular & Irregular

Aa Bb Cc Dd 0+2 4+3 8+1 12+4 2 7 9 16

Dc Cd Ba Ab 12+1 8+4 4+2 0+3 13 12 6 3

Bd Ac Db Ca 4+4 0+1 12+3 8+2 8 1 15 10

Cb Da Ad Bc 8+3 12+2 0+4 4+1 11 14 4 5

A. B. C.

132 - A regular pandiagonal magic square from a Graeco-Latin square.

The two subsidiary squares are combined in A. Values assigned (in this case) are A, B, C, D = 0, 4, 8 ,12. Lower case a, b, c, d = 2, 3, 1, 4. These values appear in square B. They are then added to give the final regular pandiagonal magic square (C).

See Solution set for a different method.

Reflection

A transformation of a magic square by exchanging the contents of cells on the right and left sides (or the top and bottom) as though the matrix was reflected in a mirror See Standard position, magic square for an illustration of rotation and reflection.

Relative frequency.

To determine the relative frequency, or degree of rareness of a magic square, cube, or hypercube, one must find all there are and divide by the number of ways of placing numbers into the array.

For example: Consider the magic cube of order 3. There are 27 numbers in the cube. This means that there are Factorial 27 = 27! = 1.088886945x1028 ways of arranging numbers.

Out of all the arrangements possible, there are 4 basic cubes, and each can be shown in 48 aspects due to rotations and reflections. This brings the total count to 192 magic cubes of order three altogether. (This is called the long count).


….. Relative frequency.

The relative frequency is then approximately 192/27! = 1.763268454x10-26.To calculate the odds against writing out a magic cube of order three, you simply find the reciprocal, which turns out to be 5.671286172x1025 : 1.

Relative frequency for some magic squares: order-3 = 8/9! = 2.204585x10 –5 ; order-5 = 275305224/25! = 1.774879092x10-18.

See Enumeration-magic squares. J. R. Hendricks, The Magic Square Course, self-published 1991.

Representation .. square

A magic square may be shown in different ways. Here are three of them. See also Intermediate and Literal squares.

aa bA Bb AB 00 13 21 32 1 8 10 15

Ab BB ba aA 31 22 10 03 14 11 5 4

bB ab AA Ba 12 01 33 20 7 2 16 9

BA Aa aB bb 23 30 02 11 12 13 3 6

A. Analytical B. Intermediate C. Conventional

133 - Three representations of an order-4 magic square.

A. A Greaco-Latin square which is obviously pandiagonal. Each symbol appears exactly 2 times in all 8 diagonals (as well as all rows and columns.

B. A magic square using quaternary (base 4) numbers. It is derived from A by assigning these values to the algebraic digits. a = 0, b = 1, B = 2 and A = 3.

C. The final pandiagonal magic square is obtained by converting the base 4 numbers to base 10 (decimal) and adding 1 to each number.

The advantage of this system is that different values may be assigned to the algebraic digits to produce different final magic squares. The value zero is used deliberately so all base 4 numbers contain two digits.

J. R.Hendricks, Inlaid Magic Squares and cubes, Self-published 1999,

0-9684700-1-7, p.7

Representation .. star … 139

Representation .. star

Magic stars have been illustrated in several pages in this book. However, diagrams take up a lot of space. Solution list may be represented concisely by showing the each solution number by number in one line of type.

The lines of the illustration are simply filled in as we trace them in order. The drawing shows the two types of patterns, continuous or separate, for all magic stars. Note that all orders, except six, have at least one continuous and, if the order number is composite, at least one separate pattern. The number of patterns per order increases for each odd order (orders 9 and 10 have 3 patterns). When you finish tracing the first separate circuit, start the next circuit at the first vacant cell after passing the top point (M in the A pattern shown here).

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

8B

A

B

C

D

E

F

G

H

I

J

K

L M

NO

P

8A

134 - Continuous and separate patterns for Order-8 magic stars.

By placing the numbers for the cell names in order we can list one solution per line


….. Representation .. star

The first 4 solutions, in index order for Order-8A

1 3 14 16 2 7 9 4 8 13 5 15 6 12 11 10

1 3 14 16 2 12 4 9 13 8 10 15 11 7 6 5

1 3 16 14 2 11 7 4 8 15 5 13 6 10 9 12

1 3 16 14 6 12 2 13 11 8 10 15 7 5 4 9

The first 4 solutions, in index order for Order-8B

A b C d e f g h i j k l m n o p

1 3 16 14 2 13 5 11 15 7 10 4 9 12 6 8

1 3 16 14 5 8 7 11 13 4 12 2 9 15 6 10

1 3 16 14 7 8 5 15 11 10 6 4 9 13 2 12

1 5 12 16 3 11 4 10 15 7 9 2 13 8 6 14

135 - First four solutions for orders 8A and 8B.

H. D. Heinz, http://www.geocities.com/~harveyh/magicstar_def.htm

12D

1

23

4

5 67

8

9

10

11

12

13

14

15

1617

18

19

20

21

22

23

24

Representation .. tesseract … 141

Representation .. tesseract

Until Hendricks published his version of a tesseract in 1962, there was no satisfactory way to visualize or diagram a magic tesseract. Researchers simply tabulated the square (or cube) arrays in the tesseract.

136 – Two traditional tesseract projections.

137 – The modern Hendricks Tesseract projection.


….. Representation .. tesseract

Fig. 136 are the two old attempts to visualize the tesseract. They cannot be partitioned which explains why Andrews, Kingery, Dr. Planck and others had difficulty visualizing it. Fig. 137 shows the modern method of depicting a magic tesseract. For order-3, the numbers are placed at the location of the dots.

Tesseracts of higher orders ( and hypercubes of higher dimension) rapidly become too complex to show in diagram form. The preferred alternative is to display them in tabular form as a series of square arrays. See Magic tesseracts for an example. Probably the only practical method for displaying these large magic figures is the one preferred by one of us (Hendricks). Simply store the hypercube in a computer program. Then, on request, print out a magic line that passes through a set of given coordinates. Or print out the coordinates for a given number. See Pathfinder.

J. R. Hendricks, Canadian Mathematical Bulletin, vol. 5, no. 2, 1962, p175

Reversible magic square

A pair of magic squares in which the digits of the numbers in one square are in reverse order to the digits in the numbers of the other square. See Ixohoxi and Upside-down magic squares where the effect depends on the symmetry of the digits used.

15 94 36 97 79 63 49 51

96 37 91 18 81 19 73 69

93 16 98 35 53 89 61 39

38 95 17 92 29 71 59 83

Original Its reverse

138 - A pair of magic squares, which have the digits in reverse order.

Reversible square … 143

Reversible square

This type of square was defined and used by K. Ollerenshaw in her work with Most-Perfect Magic Squares. While not magic, they are important because

• there is a one-to-one relationship between most-perfect and reversible squares

• the number of reversible squares of a given order may be readily determined.

Thus by simply calculating the number of reversible squares for a given order, the number of most-perfect magic squares for that order is immediately known.

Reversible squares are m x m arrays of the numbers from 1 to m2 (Ollerenshaw uses the series from 0 to m2 – 1). They have these additional properties.

The sum of the two numbers at diagonally opposite corners of any rectangle or sub-square within the reversible square will equal the sum of the two numbers of the other pair of diagonally opposite corners.

The sum or the first and last numbers in each row or column equal the sum of the next and the next to last number in each row or column, etc.

Diametrically opposed number pairs sum to m2 + 1.

1 2 3 4 1 2 3 4 5 6 7 8

5 6 7 8 9 10 11 12 1 2 3 4

9 10 11 12 5 6 7 8 13 14 15 16

13 14 15 16 13 14 15 16 9 10 11 12

A. B. C.

139 - A. Principal reversible square, B and C. 2 of its 16 variations.

These two were obtained by swapping rows (see next entry). NOTE: I have used the series from 1 to 16 in these examples to be consistent with the rest of this book. Ollerenshaw and Brée use 0 to 15.

H. D. Heinz, http://www.geocities.com/~harveyh/most-perfect.htm

K. Ollerenshaw and D. Brée, Most-Perfect Pandiagonal Magic Squares, IMA 1998, 0-905091-06-X


Reversible square, principal

Reversible squares may be assembled in sets whose members may be transformed from one to another by

Interchanging a pair of complementary rows and/or columns.

Interchanging two rows/columns in one half of the square together with interchanging the complementary rows/columns in the other half of the square.

It is therefore necessary to define which is the principle square from which the others in the set are derived from.

The principle reversible square is defined as that one containing 1 and 2 (0 and 1 if using series from 0 to m2-1) as the first two numbers in the first row and all its rows and columns have sequences of integers in ascending order.

1 2 3 4 1 2 5 6 1 2 9 10

5 6 7 8 3 4 7 8 3 4 11 12

9 10 11 12 9 10 13 14 5 6 13 14

13 14 15 16 11 12 15 16 7 8 15 16

140 - The 3 principal reversible squares of order-4.

There are three principle reversible squares for order-4, each may be transformed to 15 other reversible squares, making three sets of 16, for a total of 48 for order-4. Because each of these may be mapped to a most-perfect magic square there are 48 most-perfect magic squares for order-4. i.e. all the order-4 pandiagonal magic squares are most-perfect.

….. Reversible square, principal … 145

….. Reversible square, principal

Order

N

Principle Reversible square.

Nn

Variation of each Mn=2n-2{(1/2n)!}2

Mn

Total Most-Perfect magic squares

Nn x Mn

4 3 16 48

8 10 36864 368640

12 42 5.30842 x 108 2.22953 x 1010

16 35 2.66355 x 1013 9.32243 x 1014

32 126 4.70045 x 1035 5.92256 x 1037

141 – Number of Most-perfect magic squares K. Ollerenshaw and D. Brée, Most-Perfect Pandiagonal Magic Squares, IMA 1998, 0-905091-06-X

Right diagonal

The diagonal line of numbers from the lower left to upper right corners of the magic square.

Rotation

A transformation of a magic square (or other magic object) by rotating the magic square clockwise or counterclockwise. This produces a different aspect, (a disguised magic square). 90-degree rotations are easily accomplished using a coordinate system. For a magic square, simply replace all coordinates (x, y) by (m+1-y, x) and this square is rotated 90 degrees.

For a cube, there are four different kinds of rotation.

• Spin where (x, y, z) is replaced by (m+1-y, x, z)

• Yaw where (x. y, z) is replaced by (m+1-z, y, x)

• Roll where (x, y, z) is replaced by (x, m+1-z. y)

• Around main triagonal where (x, y, z) is replaced by (y, z, x)

See associated magic cube and basic magic cube illustrations for the rotation of a cube. See Standard position, magic square for an illustration of rotation and reflection.


Row

Each horizontal sequence of numbers. There are n rows of length n in an order n magic square.

See orthogonals for a magic cube graphic example.

11D

1

2

3

4

5

6

78

9

10

11

12

13

14

15

16

17

18

1920

21

22

S … 147

SSSS S

Indicates the magic sum. See constant.

Sagrada magic square

The Sagrada Familia cathedral in Barcelona, Spain, contains an unusual magic square.

Both the number 10 and the number 14 are repeated twice and there is no 12 or 16. The magic sum is 33. So this is not a magic square in the true sense. But what is it’s significance?

The Sagrada Familia cathedral is the most important work of Gaudi, a Spanish architect considered as a true genius. He worked on this building from 1882 until his death in 1926. Recently work has resumed in an effort to complete the building.

1 14 14 4

11 7 6 9

8 10 10 5

13 2 3 15

142 - The Sagrada magic square sums to 33.

There is some information about the cathedral at: http://www.greatbuildings.com/buildings/Sagrada_Familia.html

Two pictures of the magic square may be seen on the Heinz Web site at

http://www.geocities.com/~harveyh/unususqr.htm#Sagrada Familia


Sator word square

Magic squares are usually considered a numerical construction. However, in the middle ages when magic squares were considered amulets, and believed to have magic powers, the Sator word square was held in high esteem and believed to have magical powers. It also seemed to be of importance to the early Christian church.

This order five square is constructed from the Latin palindrome, SATOR AREPO TENET OPERA ROTAS. Because this phrase is palindromic, it reads the same backwards as forward.

S A T O R

A R E P O

T E N E T

O P E R A

R O T A S

143 - The Sator word magic square.

Self-similar magic squares

A magic square which after each number is converted to its complement, is a rotated and/or reflected copy of the original magic square. This is sometimes referred to as self-complimenting.

Mutsumi Suzuki discovered magic squares with this feature and named it self-similar. He has listed 16 order-5 magic squares and 352 order-4 magic squares of this type.

One of us (Heinz) subsequently realized that any magic square in which the complementary pairs are symmetric across either the horizontal or the vertical center line of the square is self-similar. The resulting copy is either horizontally or vertically reflected. Because associated magic squares are symmetric across both these lines, all such magic squares are self-similar and the copy is horizontally and vertically reflected from the original.

Self-similar magic squares … 149

….. Self-similar magic squares

For order-4, all 48 group III which are associated, are self-similar. Also all group VI, which are not associated but are symmetric across either the horizontal or vertical axis are self-similar.

6 26 45 9 29 18 42

40 3 23 43 11 35 20

28 48 12 31 16 36 4

17 37 1 25 49 13 33

46 14 34 19 38 2 22

30 15 39 7 27 47 10

8 32 21 41 5 24 44

144 - An order-7 associated, and thus self-similar, magic square.

If each number in this square is subtracted from 50, the same magic square emerges, but rotated 180º.The underlined cells shows the parity pattern (which is unrelated to the self-similar property).

2 7 10 15 15 10 7 2

8 12 5 9 9 5 12 8

11 1 16 6 6 16 1 11

13 14 3 4 4 3 14 13

A B

145 - An order-4 self-similar magic square that is not associated.

A. shows an order-4 that is not associated and B. shows it’s complement, which is self-similar. The process of complementing each number of a magic object is also known as ‘complementary pair interchange’ (CPI). See Heinz Self-similar Magic Squares page (self-similar.htm)

http://www.geocities.com/~harveyh/self-similar.htm

Link to Mr. Suzuki ‘s Magic Squares page from Heinz’s links page. See an excellent paper on this subject in Robert S. Sery, Magic Squares of Order-4 and

their Magic Square Loops, Journal of Recreational Mathematics, 29:4, page 274


Semi-diabolic

See Semi-pandiagonal magic square.

Semi-magic square

The rows & columns of the square sum correctly but one or both main diagonals do not. This may be generalized to n-dimensional hypercubes by saying “if one or more n-agonals do not sum correctly…”.

2332

2401 2209 961 169 25 9 1 5775

2025 16 324 4 1521 36 1849 5775

529 1369 676 484 1296 1225 196 5775

361 1681 900 625 1024 784 400 5775

289 256 225 1936 1764 576 729 5775

121 100 1089 2116 64 841 1444 5775

49 144 1600 441 81 2304 1156 5775

5775 5775 5775 5775 5775 5775 5775 7479

146 - An order-7 semi-magic square of squares.

This semi-magic square by D. M. Collison consists of the squares of the numbers from 1 to 49. See Square of squares.

Kraitchik, Maurice, Mathematical Recreations, Dover Publ., 1953, 53-9354. p. 143

J. R. Hendricks, The Magic Square Course, self-published 1991, p 23

Semi-Pandiagonal

Also known as Semi–Diabolic These magic squares have the property that the sum of the cells in the opposite short diagonals are equal to the magic constant (subject to the following conditions).

In an odd order square, these two opposite short diagonals, which together contain m-1 cells, will, when added to the center cell equal the square’s constant. The two opposite short diagonals, which together contain m+1 cells, will sum to the constant if the center cell is subtracted from their total.

Semi-pandiagonal … 151

….. Semi-Pandiagonal

In an even order square, the two opposite short diagonals which together consist of n cells will sum to the square's constant. The opposite short diagonals that together contain (3/2)m will sum to 3/2 constant., etc. Of the 880 fundamental magic squares of order 4, 384 are semi-

pan ( 48 of these are also associative).

3 16 9 22 15

2 8 15 9 20 8 21 14 2

11 13 6 4 7 25 13 1 19

14 12 3 5 24 12 5 18 6

7 1 10 16 11 4 17 10 23

A. B.

147 - Semi-pandiagonal magic squares: A. not associated, B. associated.

The underlined cells indicate the short diagonals. A. even order, opposite short diagonals = S; B. odd order, opposite short diagonals plus center = S.

Sequence patterns

The center of the cells containing consecutive numbers are joined by lines. See magic lines.

Series

Broadly speaking, series refers to the set of numbers that make up the magic object.

However, it also has a narrower meaning. A magic square usually contains m series of m numbers. The horizontal step within each series is a constant. The vertical step between corresponding numbers of each series is also a constant. This step can be but need not be the same as the horizontal step. A normal magic square has the starting number, the horizontal step and the vertical step all equal to 1.


….. Series

After the N initial series are established, the magic square is constructed using any appropriate method. If m = the squares order, a = starting number, d = the horizontal step D = the vertical step, and K = sum of numbers in the first series; then

S = (m3 + m) / 2 + m (a - 1 ) + ( K - m ) [ m ( d - 1 ) + ( D - 1 )]

See Horizontal step, Order-3 type 2, and Vertical step for examples.

W.S.Andrews, Magic Squares and Cubes,1917, pp 54-63 J.L.Fults, Magic Squares, 1974, pp 37-39

Serrated magic square

This special type of magic square was described by H. A. Sayles in The Monist sometime between 1905 and 1916 (see footnote). It is rich in unusual features, some of which are presented here.

14

19 16 30

29 24 40 8 1

39 17 5 36 11 25 22

4 27 10 7 21 15 32 35 38

20 33 31 26 37 9 3

41 18 2 34 13

12 6 23

28

148 - Order-9 serrated magic square.

….. Serrated magic square … 153

….. Serrated magic square

G.

B.

E.

F.

A.

C.

D.

149 - Patterns for the above serrated magic square

Figure A. represents the diagonal in a conventional magic square. Here it is the longest horizontal and vertical lines. B. represents the rows and columns. In this square, there are 16 of them, the same as the number of boundary cells.

The other figures also appear in this construction, but only because in this case, the embedded squares are pandiagonal. C and D each appear nine times, E and F six times, and G 12 times. You can change the order-4 pandiagonal to an associated magic square by simply exchanging rows 3 and 4, then columns 3 and 4. Make the necessary changes to the serrated square and notice the main features are still valid but figures C to G no longer are.

14 30 1 22 38

16 8 25 35 19 40 11 32 3

24 36 15 9 29 5 21 37 13

17 7 26 34 39 10 31 2 23

27 33 18 6 4 20 41 12 28

150 - Two pandiagonal magic squares contained in the above serrated magic square.

Here we show these two magic squares in the more normal orientation. Rotate 45º clockwise to see how they fit into the serrated square.

W.S.Andrews, Magic Squares and Cubes, 1917, pp241-244


Short diagonal

One which runs parallel to a main diagonal from 1 side of the square to an adjacent side. For a magic square of order 2m, each short diagonal contains m cells. For a magic square of order 2m-1, each short diagonal contains (m-1)/2 cells. See Semi-pandiagonal magic squares and Opposite short diagonals for illustrations.

Used by some authors on magic cubes to mean the diagonals of a square face, or cross section of a cube. For this case, Hendricks uses diagonal, or 2-agonal instead.

See also Long Diagonal.

Simple Magic Square

A square array of numbers, usually integers, in which all the rows, columns, and the two main diagonals have the same sum. As these are the minimum specifications to qualify as a magic square this term signifies it has no special features. The one order 3 magic square is not simple (it is associative). Of the 880 order 4 magic squares, 448 are classified as simple. A broad classification of magic squares is;

Simple

Associated

Semi-pandiagonal

Pandiagonal (perfect)

Combinations; Associated with Semi-pandiagonal or pandiagonal

Singly-even order

The side of the square is divisible by two but not by four. This is the most difficult order to construct. Order-6 is the smallest singly-even order magic square.

Also called oddly-even order.

….. Singly-even … 155

….. Singly-even order

6 7 26 27 22 23

8 5 28 25 24 21

34 35 18 19 2 3

36 33 17 20 4 1

14 15 10 11 30 31

13 16 12 9 29 32

151 - An order-6 simple, normal, singly-even magic square

It is not possible to have an order-6 magic square consisting of the numbers 1 to 36 (or any other consecutive numbers) that is also pandiagonal or associated.

W. S. Andrews, Magic Squares and Cubes, Dover, Publ., 1960, p.266.

Skew related

More modern terms are Symmetrical cells and Diametrically equidistant.

RouseBall & Coxeter, Mathematical Recreations and Essays, 1892, (13 Edition, p.194)

12C

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Solution set

The set of numbers assigned to algebraic symbols which may bring about the solution one seeks. For example, one seeks a pandiagonal magic square of order 6. One devises a pattern, as follows.

bA cC aB BA CC AB

aC bB cA AC BB CA

cB aA bC CB AA BC

ba cc ab Ba Cc Ab

ac bb ca Ac Bb Ca

cb aa bc Cb Aa Bc

This pattern shows a general pandiagonal magic square of order six. a,A,b,B,c,C stand for 0,1,2,3,4,5 in some order or other.

There is a provision that a+A=b+B=c+C=m-1. If you study the lines of first digits and the lines of second digits, you will find that in the diagonals, they will all sum the same sum which would be 15 from. the

152 - An algebraic pattern for an order-6 pandiagonal magic square.

In the first column, b+a+c+b+a+c must sum 15, which is impossible. Therefore, in the number system based six, although one can get the diagonals to sum a magic sum, we cannot get the rows and columns to do so. The goal is to obtain a pandiagonal magic square, not necessarily a normal one. So, what we do is increase the order m to 7, or any odd number higher than 7 and get a solution.

Using the number system base 7.with digits 0,1,2,3,4,5,6 , if we omit the 3 and use the others, there is a solution. The equation becomes: a+A=b+B=c+C=7-1=6. Below are one solution (you can find others), one magic square in the number system base 7 and its conversion to the decimal number system.

Solution set: a = 0, A = 6, b = 4, B = 2, c = 5, C = 1

….. Solution set … 157

…... Solution set

46 51 02 26 11 62 35 37 3 21 9 45

01 42 56 61 22 16 2 31 42 44 17 14

52 06 41 12 66 21 38 7 30 10 49 16

40 55 04 20 15 64 29 41 5 15 13 47

05 44 50 65 24 10 6 33 36 48 19 8

54 00 45 14 60 25 40 1 34 12 43 20

Pandiagonal – base 7, then plus 1 to Pandiagonal – base 10

153 - The order-6 pandiagonal in base 7 and base-10

And don’t be surprised if you get a bonus, The base ten square sums 150 and it is bimagic in rows and columns with the sum 5150.

See Intermediate square and Literal square.

10A

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1920


Space diagonal

A line joining opposite corners of a hypercube. When moving along the line, all n coordinates will change (n is dimension of the hypercube).

See triagonals, quadragonals and n-agonals.

Species, order-3

Species is a consideration of the placement of even and odd numbers in the normal order-3 magic square, cube and tesseract. This classification has no meaning if the magic square, etc. does not consist of consecutive numbers. For instance, an order-3 prime number magic square must consist of all odd numbers.

154 - Even and odd number placement in a magic square and cube.

There is only one basic magic square of order-3, and so only 1 species with the even numbers appearing on the four corners. The magic cube must have three even numbers on two edges of each of the six faces. So, even though there are four basic order-3 magic cubes, there is only one species.

An even number

An odd number

….. Species, order-3 … 159

….. Species, order-3

155 - Species # 1 of the order-3 magic tesseract.

There are three types of order-3 magic tesseracts. In each case, the type of species is determined by the even and odd numbers in the lines (row, column, pillar, file) radiating from corners that have odd numbers. Species # 1; All four lines consist of two even numbers (plus the odd corner number). Of the 58 basic magic tesseracts, only 2 are of this species. Species # 2; two lines have all odd numbers, the other two lines have 2 even numbers (plus the odd corner number). 24 basic tesseracts are of this species. Species # 3; Through an odd corner numbers, either the other two numbers are even in only one line, or odd in only one line. The remaining 32 basic tesseracts belong to this species.

J. R. Hendricks, Species of Third Order Magic Squares and Cubes, JRM 6:3, 1973, pp.190-192.

J. R. Hendricks, All Third Order Magic Tesseracts, self-published 1999,

0-9684700-2-5, pp 4-7.


Square of squares.

This is a Number Square where when you square the numbers it becomes magic. It is important because you have four types:

Number --- not magic

Semi-magic --- only rows and columns sum correctly

Magic -- sums a constant in first degree

Square of Squares - sums a constant second degree

Bimagic - sums a constant in either degree

Nobody yet has determined if it is possible for a square of squares to be fully magic (when the original numbers do NOT form a magic square).

Following are two examples of a semi-magic square of squares. Kevin Brown calls these Orthomagic squares of squares. Brown shows proof on his Web site that an order-3 of this type cannot be magic.

See also Semi-magic for an order-7 example.

155 8571

11 23 71 105 121 529 5041 5691

61 41 17 119 3721 1681 289 5691

43 59 19 121 1849 3481 361 5691

115 123 107 71 5691 5691 5691 2163

A. B

156 - Square of prime numbers make a semi-magic square.

A. Original numbers (all primes) but not magic.

B. Square of these prime numbers form a semi-magic square.

….. Square of squares … 161

….. Square of squares.

143 6849

4 23 52 79 16 529 2704 3249

32 44 17 93 1024 1936 289 3249

47 28 16 91 2209 784 256 3249

83 95 85 64 3249 3249 3249 2208

A. B.

157 - The smallest orthomagic arrangement of distinct squares.

A. Original number square.

B. Squares of numbers. The rows and columns all sum to 3249, the square of 57.

From Kevin Browns Web site at http://www.seanet.com/~ksbrown/kmath427.htm

Sringmagic

An array of m cells in the shape of a small ring that appears in each quadrant of an order-n quadrant magic square. This is one of the first 5 patterns discovered. However, This pattern doesn’t exist for order-5.

See Quadrant magic patterns and Quadrant magic square.

158 - Sringmagic quadrant pattern for order-9 and order-13


Standard position - magic square

Any magic square may be disguised to make 7 other (apparently) different magic squares by means of rotations and/or reflections. These variations are NOT considered as new magic squares for purposes of enumeration. For the purpose of listing and indexing magic squares, a standard position must be defined. The magic square is then rotated and/or reflected until it is in this position. This position was defined by Frénicle in 1693 and consists of only two requirements.

Conditions for standard position:

The lowest of any corner number must be in the upper left hand corner.

The cell in the top row adjacent to the top left corner must be lower then the leftmost position of the second row (also adjacent to the top left corner).

This process is called Normalizing. Achieving the first condition may require rotation. The second may require rotation and reflection. Once the magic square is in this position, it may be put in the correct index position in a list of magic squares of a given order.

This definition has meaning (and relevance) only for a normal

magic square.

Another term often used for a magic square with these qualifications is Fundamental.

4 9 6 15 5 16 3 10 5 14 11 4

11 8 13 2 14 1 12 7 16 1 8 9

14 1 12 7 11 8 13 2 3 12 13 6

5 16 3 10 4 9 6 15 10 7 2 15

A. Standard position B. C.

159 – Order-4, standard position, and two disguises.

Magic square A. is the basic version; B. copy but vertically reflected, C copy but rotated 90º clockwise.

Bensen & Jacoby, New Recreations with Magic Squares, 1976, p 123.

Standard position – magic star … 163

Standard position - magic star

A magic star may be disguised to make 2n-1 apparently different magic stars where n is the order (number of points) of the magic star.

Three characteristics determine the Standard position. � The diagram is oriented so only one point is at the top. � The top point of the diagram has the lowest value of all the points. � The valley to the right of the top point has a lower value then that of the valley to the left.

This process is called Normalizing. Achieving the first and second conditions may require rotation. The third may require reflection. Once the magic star is in this position, it may be put in the correct index position in a list of magic stars of a given order. This definition has meaning (and relevance) only for a normal magic star. See the Heinz web page on Magic Star Definitions.

160 - An order-6 magic star and a disguised version of it.

Star A. is index # 23 of the basic 80 order-6 magic stars. It is in the standard position because 1 is the smallest point number and it is at the top. The second condition is that the 6 in the valley to the right of the top point is smaller then the 9 in the valley to the left.. Star B. is a disguised version of star A. To be put in the standard position (normalized) it must first be rotated one position clockwise, then it has to be reflected horizontally.

H. D. Heinz, http://www.geocities.com/~harveyh/magicstar_def.htm

1

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74105

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3 1

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5119

8 2

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A. B.


Subtraction magic square

Interchange the contents of diagonal opposite corners of an order-3 addition magic square. Now, if you add the two outside numbers and subtract the center one from the sum, you get the constant 5.

The Division magic square is a similar conversion from the Multiply (Geometric) magic square.

8 1 6 2 1 4

3 5 7 3 5 7

4 9 2 6 9 8

A B

161 - Subtraction magic square

A. order-3 magic square, B. resulting subtract magic square. QB = 5

Summations

The magic sum for an n-Dimensional Magic Hypercube of Order m is given by:

S = m(1 + mn)/2

In a magic object, there are many lines that produce the magic sum. The table below, shows the minimum requirement of the number of lines for various types of magic hypercubes and is derived from the following equation:

N = 2(r-1)n!m(n-1)/[r!(n-r)!]

• Where: N is the number of r-agonals

• n is the dimension of the hypercube

• m is the order of the hypercube, and

• r is the dimension of the hyperplane.

When r = 1, the number of orthogonals is given by N. As well, shown is the smallest order for the various classifications of pandiagonal, pantriagonal, etc. which is known. for each dimension. Some of the tesseracts are not known yet and some of these varieties have not been constructed yet.

….. Summations … 165

….. Summations

This table provides the minimum requirements for each category. Usually, there are some extra lines which may sum the magic sum, but not a complete set so as to change the category.

Magic Lowest i-row n-agonals

Hypercube Order 1 2 3 4 Total

Square

Regular 3 2m 2 2m + 2

Pandiagonal 4 2m 2m 4m

Cube

Regular 3 3m2 4 3m2 + 4

Diagonal 5 3m2 6m 4 3m2+6m+4

Pantriagonal 4 3m2 4m2 7m2

PantriagDiag 8? 3m2 6m 4 m2 7m2+6m

Pandiagonal 7 3m2 6m2 4 9m2 + 4

Perfect 8 3m2 6m2 4m2 13m2

Tesseract

Regular 3 4m3 8 4m3 + 8

Pandiagonal ? 4m3 12m3 8 16m3 + 8

Pantriagonal ? 4m3 16m3 8 20m3 + 8

Panquadrag-onal

4 4m3 8m3 12m3

Pan2 + Pan3 ? 4m3 12m3 16m3 8 32m3 + 8

Pan2 +Pan4 ? 4m3 12m3 8m3 24m3

Pan3 + Pan4 ? 4m3 16m3 8m3 28m3

Perfect 16 4m3 12m3 16m3 8m3 40m3

162 - Hypercubes – number of correct summations.

Symmetrical cells

Two cells that are the same distance and on opposite sides of the center of the cell are called symmetrical cells. In an odd order square the center is itself a cell. In an even order square the center is the intersection of 4 cells. Other definitions for these pairs are skew related and diametrically equidistant (illustration).


….. Symmetrical cells

X

2 Y Z

Z X Y

Y X Z Z Y

1 2 1 1/2 X 1

163 - Symmetrical cells in even and odd order magic squares.

X, Y and Z in each case are symmetrical cells. 1 is symmetrical around the vertical axis only and 2 around the horizontal axis only.


RouseBall & Coxeter, Mathematical Recreations and Essays,1892 (13 Edition,

p.194,202)

Symmetrical magic square

See Associated Magic Square.

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Talisman magic square … 167

TTTT Talisman magic square

A Talisman square is an m x m array of the integers from 1 to m2 so that the difference (D) between any integer and its neighbors, horizontally, vertically, of diagonally, is greater then some given constant. The rows, columns and diagonals will NOT sum to the same value so the square is not magic in the normal sense of the word. This type of square was discovered and named by Sidney Kravitz.

28 10 31 13 34 16

15 1 12 4 9 19 1 22 4 25 7

20 7 22 18 24 29 11 32 14 35 17

16 2 13 5 10 20 2 23 5 26 8

21 8 23 19 25 30 12 33 15 36 18

17 3 14 6 11 21 3 24 6 27 9

D>4 D>8

164 - Two Kravitz Talisman squares Joseph S. Madachy, Mathemaics On Vacation, 1966, pp 110-112.

Tesseract

A four-dimensional equivalent to a cube. A regular four-dimensional hypercube. It is bounded by 16 corners, 32 edges, 24 squares, 8 cubes. See Basic magic tesseract, Magic

tesseract, Partitioning, Perfect magic tesseract, and Quadragonal for illustrations.

Pickover, Clifford A., The Zen of Magic Squares, Circles and Stars, Princeton Univ.

Press, 2002, 0-691-07041-5, page 117


Transformation

Any order-5 pandiagonal magic square may be converted to another magic square by permuting the rows and columns in the order 1-3-5-2-4. Each of these two magic squares can be transformed to another by exchanging the rows and columns with the diagonals. Finally, each of these four squares may be converted to 24 other magic squares by moving one row (or column) at a time to the opposite side. See cyclical permutations.

Any order-5 magic square can be transposed to another one by either of the following two transformations.

• Exchange the left and right columns, then the top and bottom rows.

• Exchange columns 1 and 2 and columns 4 and 5. Then exchange rows 1 and 2, and rows 4 and 5.

These methods, of course, also work for all odd orders greater then order-5.

Another type of transformation converts any magic square to its compliment by subtracting each integer in the magic square from n2 + 1. In some cases this results in a copy of the original magic square. Still other types of transformations involve complementing digits of the numbers when represented in the radix of the magic square order.

See the Heinz Transformation pages which shows more then 45 transformations (for order-4).

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165 - Transforming an associated magic square to a pandiagonal magic square by changing quadrants to rows. H. D. Heinz, http://www.geocities.com/~harveyh/transform.htm

Benson & Jacoby, Magic Squares & Cubes, 1976, pp.128-131.

Translocation … 169

Translocation

When the left column of numbers is moved to the right side of the magic square, or vice-versa. Or the top row is moved to the bottom (or the bottom to the top). For cubes, the front face may be moved to the back, etc. This works for any dimension of hypercube, but only if the figure is pan-n-agonal or perfect. If we tried this with a pan-diagonal magic cube, it would change the triagonals and the new triagonals may not sum correctly.

In a pan-4-agonal magic tesseract, an entire facial cube may be shifted to the other side of the tesseract and it remains magic. This is because we place the emphasis that the main n-agonals must sum the magic sum.

This is a type of Transformation.

Transposition

The permutation of the rows and columns of a pandiagonal magic square in order to change it into another pandiagonal magic square.

For order-5 this is cyclical 1-3-5-2-4. For order-7 there are two non-cyclical permutations, 1-3-5-7-2-4-6 and 1-4-7-3-6-2-5.

Another transposition method for pandiagonals is to exchange the rows and columns with the diagonals.

Benson & Jacoby, Magic squares & Cubes, Dover 1976, 0-486-23236-0, pp.146-154.

The above authors devote a chapter in their book to transposition, but freely use the term transformation elsewhere in the same book. Other authors seem to prefer the term transformation. In general, either term may be considered any method of converting one magic square into another one.

Traditional magic square

See Magic square, normal


Triagonal

A space diagonal that goes from 1 corner of a magic cube to the opposite corner, passing through the center of the cube. There are 4 of these in a magic cube and all must sum correctly (as well as the rows, columns and pillars) for the cube to be magic. As you go from cell to cell along the line, all three coordinates change. In tesseracts this is called a quadragonal. For higher order hypercubes, this is called an n-agonal or space diagonal. Of course, with these higher dimensions there are more coordinates. A triagonal is sometimes called a long diagonal. See orthogonal for an illustration.

Triagonals in one direction

Order 1 segments 2 segments 3 segments Total

3 1 6 2 9

4 1 9 6 16

5 1 12 12 25

6 1 15 20 36

7 1 18 30 49

8 1 21 42 64

9 1 24 56 81

10 1 27 72 100

166 – # of parallel segmented triagonals for orders 3 to 10

Because there are four triagonals in a magic cube, the above figures must be multiplied by four to obtain the actual number of triagonals in the cube.

J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999.

Trimagic Square

Also called Triplemagic. A magic square in which all the number lines sum correctly, when each number is squared the lines sum correctly, and when each number is cubed the lines sum correctly. Benson and Jacoby show a method to produce order-32 trimagic squares, the smallest so far constructed. See Bimagic Square for an illustration.

Benson & Jacoby, Magic squares & Cubes, Dover 1976, 0-486-23236-0

Upside-down magic square … 171

UUUU Upside-down magic square.

The digits 0, 1 and 8 have horizontal and vertical symmetry and so read the same right-side up, in reverse, and upside-down. See Ixohoxi magic square for an order-8 square of this type that uses digits 1 and 8 only. The 6 and the 9 may be added to this list, but in their case the upside-down 6 becomes a nine. and the upside down 9 a six.

The upside-down magic square below is produced using only these five digits. When it is turned upside down, by 180º rotation, a new magic square is produced. The square may also be viewed upside down by reflection. This produces still another magic square, but in this case digits 6 and the 9 are reversed.

Of course, in all cases, the resulting magic square is only a disguised version of the original. The novelty is due to the fact that the numbers read correctly. See Ixohoxi. See also Reversible magic square, which doesn’t depend on symmetrical digits.

167 - Mr. Collison’s order-5 pandiagonal upside-down magic square.

J. R. Hendricks, The Magic Square Course, self-published 1991, p 31

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VVVV Vertical step

The difference between corresponding numbers of the m series. It is not a reference to the rows of the magic square.In a normal magic square, the horizontal step and vertical step are both 1.

1480028201 1480028129 1480028183

1480028153 1480028171 1480028189

1480028159 1480028213 1480028141

168 - The smallest possible consecutive prime magic square.

In fig. 168, the horizontal step is 12 and the vertical step is 6. Because the vertical step is positive, this is a type 1 magic square. It was discovered, along with 21 others, by Harry L. Nelson in 1988.

23813359751 23813359613 23813359727

23813359673 23813359697 23813359721

23813359667 23813359781 23813359643

169 - The smallest possible Type 2 consecutive prime magic square.

This magic square consists of three triplets of primes with horizontal step of 30 and vertical step of –6. It was recognized by Aale de Winkel in August, 1999 as a Type 2 magic square, working from a list of consecutive prime number sequences found by Harry L. Nelson in 1988. See Horizontal step and Order-3, type 2.


W.S.Andrews, Magic Squares and Cubes,1917

E-mail messages between Aale de Winkel, H. D. Heinz, and Harry J. Smith in July and August, 1999.

Harry L. Nelson, J. Recreational Mathematics, 20:3, 1988, A Consecutive Prime 3 x

3 Magic Square. H. D. Heinz, http://www.geocities.com/~harveyh/type2.htm

Weakly-magic stars … 173

W Weakly- magic stars

Marián Trenkler of Safarik University, Slovakia, refers to a magic star that does not use consecutive numbers (i.e. not normal) as weakly-magic.

170 - An order-8B star that is weakly magic because numbers are not consecutive.

H. D. Heinz, http://www.geocities.com/~harveyh/trenkler.htm

Marián Trenkler, Magicke Hviezdy (Magic stars), Obsory Matematiky, Fyziky a

Informatiky, 51(1998).

9

11

8

72

12

3

16

18

27

292531

37 34

35

SW

8


Wrap-around

Used in pandiagonal magic squares to indicate that lines are actually loops. Each edge may be considered to be joined to the opposite edge. If you move from left to right along a row, when you reach the right edge of the magic square, you wrap-around to the first cell on the left.

Or consider that the pandiagonal magic square is repeated in all four directions. Any n x n section of this array may be considered as a pandiagonal magic square. This results from the fact the broken diagonal pairs form complete lines.

See Broken-diagonal pair for an illustration.

1 2 3 4 5 1 2 3 4 5 1 2 3 4

6 7 8 9 10 6 7 8 9 10 6 7 8 9

11 12 13 14 15 11 12 13 14 15 11 12 13 14

16 17 18 19 20 16 17 18 19 20 16 17 18 19

21 22 23 24 25 21 22 23 24 25 21 22 23 24

1 2 3 4 5 1 2 3 4 5 1 2 3 4

6 7 8 9 10 6 7 8 9 10 6 7 8 9

11 12 13 14 15 11 12 13 14 15 11 12 13 14

16 17 18 19 20 16 17 18 19 20 16 17 18 19

21 22 23 24 25 21 22 23 24 25 21 22 23 24

1 2 3 4 5 1 2 3 4 5 1 2 3 4

6 7 8 9 10 6 7 8 9 10 6 7 8 9

11 12 13 14 15 11 12 13 14 15 11 12 13 14

171 - An unorthodox use of wrap-around generates an order-5 magic square.

This unconventional use of wrap-around may be used to generate odd order magic squares. Broken diagonal pairs better illustrates the normal meaning of the term.

J. R. Hendricks, The Magic Square Course, self-published 1991, p 67.

References 175

David H. Ahl, Computers in Mathematics: A Sourcebook of Ideas. Creative Computer Press,1979, 0-916688-16-X, P. 117 W. S. Andrews, Magic Squares and Cubes, Dover, Publ., 1960, pp124-125 Benson & Jacoby, Magic squares & Cubes, Dover 1976, 0-486-23236-0 Kevin Brown, http://www.seanet.com/~ksbrown/kmath427.htm Kevin Brown, http://www.seanet.com/~ksbrown/kmath353.htm L. E. Card, J. Recreational Mathematics, 1:2, 1968, pp.93-99. Cormie & Linek’s anti-magic square page at http://www.uwinnipeg.ca/~vlinek/jcormie/index.html H. E. Dudeney, Amusements in Mathematics,Dover 1958, 0-486-20473-1 (Reprint of 1917 work) J. L. Fults, Magic Squares, Open Court 1974, 0-87548-197-3 Martin Gardner, New Mathematical Diversions from Scientific American, Simon & Schuster 1966, 66-26153. pp.162-172.

176

H. D. Heinz web page on Iso-like Magic Stars http://www.geocities.com/~harveyh/panmagic.htm H. D. Heinz web page on Magic Star Definitions http://www.geocities.com/~harveyh/magicstar_def.htm H. D. Heinz web page, Most-Perfect Magic Squares http://www.geocities.com/~harveyh/most-perfect.htm H. D. Heinz web page, Quadrant Magic Squares http://www.geocities.com/~harveyh/quadrant.htm H. D. Heinz web page, Self-similar Magic Squares http://www.geocities.com/~harveyh/self-similar.htm H. D. Heinz web page, Transformations http://www.geocities.com/~harveyh/transform.htm H. D. Heinz web page, Trenkler Stars http://www.geocities.com/~harveyh/trenkler.htm H. D. Heinz Web page on 3-D Magic Stars http://www.geocities.com/~harveyh/3-d_star.htm H. D. Heinz Web page on Tree-planting graphs. http://www.geocities.com/~harveyh/order5.htm H. D. Heinz Web page on Unusual Magic Stars. http://www.geocities.com/~harveyh/unusualstr.htm H. D. Heinz Web page on Order-3 type 2. http://www.geocities.com/~harveyh/type2.htm

177

H. D. Heinz Web page on Prime Magic Stars. http://www.geocities.com/~harveyh/primestars.htm J. R. Hendricks, All Third Order Magic Tesseracts, self-published 1999, 0-9684700-2-5 J. R. Hendricks, Bimagic Squares: Order 9, self-published 1999, 0-9684700-6-8 J. R. Hendricks, A Bimagic Cube Order 25, self-published 1999, 0-9684700-6-8 and Danielsson, Printout of A Bimagic Cube Order 25, 2000 J. R. Hendricks, Inlaid Magic Squares and Cubes, Self-published 2000, 0-9684700-7-6 J. R. Hendricks, The Magic Square Course, self-published 1991, p 32 J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999, 0-9684700-0-9 J. R. Hendricks, Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published,1999, 0-9684700-4-1 J. R. Hendricks, American Mathematical Monthly, Vol. 75, No. 4, April 1968, p.384 J. R. Hendricks, Canadian Mathematical Bulletin, Vol. 5, No. 2, 1962, p175

J. R. Hendricks, J. Recreational Mathematics, 25:4, 1993, pp 286-288, An Inlaid Magic Cube

178

A. W. Johnson, Jr., J. Recreational Mathematics 15:2, 1982-83, p. 84 Katagiri & Kobayashi, J. Recreational Mathematics, 15:3, 1982-83, pp200-208, Magic Triangular Regions of Orders 5 and 6. M. Kraitchik, Mathematical Recreations., Dover Publ. , 1942, 53-9354, pp 166-170 Joseph S. Madachy, Mathematics On Vacation, Nelson, 1966, 17-147099-0 Jim Moran Magic Squares, 1981, 0-394-74798-4 Harry L. Nelson, J. Recreational Mathematics, 20:3, 1988, p.214-216. Ollerenshaw and D. Brée, Most-Perfect Pandiagonal Magic Squares, IMA 1998, 0-905091-06-X Carlos Rivera’ Prime Problems & Puzzles WWW site http://www.primepuzzles.net/ RouseBall & Coxeter, Mathematical Recreations and Essays, 1892, 13 Edition, p.194 Lee SallowsAbacus 4, 1986, pp.28-45 & 1987 pp.20-29 Harry J. Smith at http://home.netcom.com/~hjsmith/ Ian Stewart, Mathematical Recreations column in Scientific American, November 1999

179

Mutsumi Suzuki’s large WWW site now at http://mathforum.org/te/exchange/hosted/suzuki/MagicSquare.html Marián Trenkler, Obzory Matematiky, Fyziky a Informatiky, 1998, no. 51, pp.1-7, Magic Stars Marián Trenkler, The Mathematical Gazette March 2000, A Construction of Magic Cubes. C. Trigg, J. Recreational Mathematics, 10:3, 1977, pp 169-173, Anti-magic pentagrams. C. W. Trigg, J. Recreational Mathematics, 11:2, 1984-85, pp.105-107, Perimeter Anti-magic tetrahedrons and Octahedrons. C. W. Trigg, J. Recreational Mathematics, 17:2, 1978-79, pp.112-118, Nine-digit Digit-root Magic Squares. C. W. Trigg, J. Recreational Mathematics, 29:1, 1998, pp.8-11, Almost Magic Pentagams Terrel Trotter, Jr., J. Recreational Mathematics, 7:1, 1974, pp.14-20, Perimeter-magic Polygons. Usiskin & Stephanides, J. Recreational Mathematics, 11:3, 1978-79, pp.176-179, Magic Triangular Regions of Orders

4 and 5. Aale de Winkels WWW site on magic subjects at http://www.adworks.myweb.nl/Magic/

180

Some additional Web sites with material on magic squares. Suzanne Alejandre, magic squares for math education http://forum.swarthmore.edu/alejzndre/magic.square.html Holgar Danielsson's Magic Squares at http://www.magic-square.de Bogdan Golunski’s big magic squares at www.golunski.de/ Alan Grogono’s Magic Squares by “Grog” http://www.grogono.com/magic/ Meredith Houlton’s WWW site www.inetworld.net/~houlton/ Fabrizio Pivari’s strange magic squares WWW site at www.geocities.com/CapeCanaveral/Lab/3469/ ml F. Poyo magic Squares, Cubes & Hypercubes http://makoto.mattolab.kanazawa-it.ac.jp/~poyo/magic/ Kwon Young Shin http://user.chollian.net/~brainstm/MagicSquare.htm R. C. Wilke Nested magic squares http://members.aol.com/robertw653/magicsqr.html

181

The Authors

Harvey D. Heinz Harvey Heinz was born in Edmonton, Alberta, Canada, the oldest of 5 boys and one girl, and moved to Vancouver, British Columbia at age 10. He entered the printing industry at age 15 as an apprentice paper ruler. At that time, unknown to him, it was already a dying industry. On his semi-retirement in 1991, he was the only paperuler still operating from Toronto west, and probably the only one in all of Canada. He was always interested in mathematical puzzles and especially number patterns. He was also interested in electronics and became an amateur radio operator in 1948, building all his own equipment. This evolved to where he was building radio controlled model boats (at a time when all equipment had to be home constructed). This in turn changed to an interest in building simple game machines, thus combining his interests in electronics and mathematical logic. These little machines were entered in local hobby shows under the naïve name of intelligent machines. In 1956 Harvey married Erna Goerz and they subsequently had two sons, Randal and Gerald. It was about this time that computers and robots were coming onto the scene and he started devouring everything he could find on these subjects in the popular press. Of course, all this time he was still collecting puzzles and number patterns. In 1958 he designed EDRECO, (Educational RElay Computer), and after obtaining about 5 tons of obsolete equipment from the local telephone company, started a computer club of senior high school students. The club started with about 25 members, but after three years, when it disbanded, had dwindled down to three (all original) members. By this time several units of the computer were built and operating successfully, the most notable being the arithmetic logic unit (ALU).

182

In 1973, Harvey was suddenly out of a job, so decided to work part-time at his trade and concentrate on bringing some of his electronic games to market. During this time, he attended many free engineering seminars on computer circuits (which were actually available to anyone) that were put on by the new semi-conductor manufacturers as a selling ploy. He also took several technical courses at the local Institute of Technology by brazenly writing prerequisite exams.

By 1977, he realized his plans were not practical so he and wife Erna started a printer trade bookbindery. By 1983 sons Randy and Gerry were both involved with the company and it was starting to grow. At that time the boys bought a half interest in the company so they could participate in this growth. In 1991 Harvey and Erna sold them the other half interest and semi-retired.

Now Harvey had time to get back to his hobbies. Building electronic hardware was now replaced by operating computers. This fit in perfect with his interest in number patterns! He was now able to investigate all sorts of patterns that previously he had just wondered about.

Heinz’s major accomplishments in number patterns.

• Found all solutions for magic stars orders 6 to 11 (by computer exhaustion).

• Found all solutions for order-12 pattern B magic stars (826,112) and most for the other 3 patterns of this order.

• Found all minimal and smallest consecutive primes solutions for orders 5 and 6 prime number magic stars.

• Discovered a 3-D magic star (in association with Aale de Winkel).

• Investigated Isolike, Pan-magic stars and Quadrant magic squares also in association with Aale de Winkel).

• Investigated Self-similar magic squares.

• Publishes a large Web site on number patterns at http://www.geocities.com/~harveyh/

183

The Authors

John R. Hendricks

Mr. Hendricks worked for the Canadian Meteorological Service for 33 years and retired in 1984. At the beginning of his career, he was a NATO training instructor. He worked at various forecast offices in Canada and eventually became a supervisor. Throughout his career, he was known for his many contributions to statistics and to climatology.

While employed, he also participated in volunteer service groups. He was Chairman, Manitoba Branch and earlier Saskatchewan Branch, The Monarchist League of Canada.

He was appointed by the Lieutenant-Governor of Manitoba as President, Manitoba Provincial Council, The Duke of Edinburgh’s Award in Canada. He was also appointed by the Governor General of Canada to the National Council of The Award Program.

Later, he was conferred with the Canada 125 Medal for his volunteer work.

.John Hendricks started collecting magic squares and cubes when he was 13 years old. This became a hobby with him and eventually an obsession. He never thought that he would ever do anything with it. But soon, he became the first person in the world to successfully make and publish four, five and six-dimensional magic hypercubes. He also became the first person to make inlaid magic cubes and a wide variety of inlaid magic squares. He has written prolifically on the subject in the Journal of Recreational Mathematics.

184

Several major discoveries he made within the last two years are:

• an inlaid magic tesseract.

• the placement of numbers for a perfect magic tesseract of order 16.

• the placement of numbers in a perfect five-dimensional magic hypercube of order 32

• a new method of doubling the order of a given square, cube, or tesseract.

• a new method of making bimagic squares of order nine.

• the world’s first bimagic cube of order 25.

During his retirement, he has also:

• Given many public lectures on magic squares.

• Given many lectures to teachers at in-service sessions.

• Developed a magic square course for the gifted junior high students.

• Delivered half a dozen colloquia to professors.

• Assisted with the Shad Valley program for young people.

• Written many books on the subject.

Appendix A1-1

MAGIC SQUARE BIBLIOGRAPHY

The following bibliography consists of books, chapters from books, and articles published during the 20th century, that deal with magic squares, cubes, stars, etc. Because it contain only material that I am personally acquainted with (except for those mentioned on this page), it is obviously not complete. However, it does contain more than 140 items.

H. D. Heinz For 18th and 19th century books on the subject see Early Books on Magic Squares, W. L. Schaaf, JRM:16:1:1983-84:1-6 Some books on magic squares published prior to that time are Agrippa De Occulta Philosophia (II, 42) 1510 Bachet Problems plaisans et delectables 1624 Prestet Nouveaux Elemens des

Matématiques 1689

De la Loubere Relation du Royaume de Siam 1693 Frenicle Des Quarrez Magiques. Acad. R.

des Sciences 1693

Ozonam Récréations Mathématiques 1697

From

Falkener, Edward, Games Ancient and Oriental, Dover Publ., 1961, 0-486-20739-0.

MAGIC SQUARE BIBLIOGRAPHY Page A1-2

The following books are wholly concerned with magic squares (and related subjects).

Andrews, W.S., Magic Squares & Cubes, Dover Publ., 1960 (original publication Open Court,1917) This book seems to be the definitive text on magic squares. The essays which comprise this volume appeared first in an American journal called The Monist between 1905 and 1916 and were written by different authors.

Benson, W. & Jacoby, O., New Recreations with Magic Squares, Dover Publ., 1976, 0-486-23236-0 This book is a serious attempt to bring the theory of magic squares up to date (1976). The authors present a new method of cyclically developing magic squares. They include a listing of all 880 4 by 4 magic squares. A chapter shows how to generate all 3600 5x5 pandiagonal magic squares.

Benson, W. & Jacoby, O., Magic Cubes: New Recreations, , Dover Publ., 1981, 0-486-24140-8 This book provides a valuable contribution to the literature, including the first(?) (not by new definition) perfect order-8 magic cube..

Candy, A. L. Pandiagonal Magic Squares of Prime Order, self-published 1940. A small hard-bound book with much theory on this subject.

Fults, John Lee, Magic Squares, Open Court Publ., 1974, 0-87548-197-3 This book contains a wealth of information on all types of magic squares. It is written as a text book and includes exercises at the end of each chapter.

Hendricks, John R., The Magic Square Course., Unpublished, 1991, 554 pages 8.5 “ x 11” binding posts. Written for a high school math enrichment class he conducted for 5 years


Hendricks, John R., A Magic cube of Order-10, Unpublished, 1998, 23 pages 8.5 “ x 11” flat stitched. …With an inlaid cube of order-6 and adorned with 12 inlaid magic squares of order-6.

Hendricks, John R., Magic Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9 212 pages plus covers, 8.5” x 11” spirol bound, 100+ diagrams. Lots of theory and diagrams, new methods and computer programs. 3 appendices.

Hendricks, John R., Inlaid Magic Squares and Cubes, Self-published, 1999, 0-9684700-1-7 206 pages plus covers, 8.5” x 11” spiral bound, 100+ diagrams. Lots of theory and diagrams. Includes a list of 46 mathematical articles published in periodicals by the author.

Hendricks, John R., All Third-Order Magic Tesseracts, Self-published, 1999, 0-9684700-2-5 36 pages plus covers, 8.5” x 11” flat stitched, 60+ diagrams. Some theory. Lots of diagrams.

Hendricks, John R., Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published, 1999,

0-9684700-4-1. 36 pages plus covers, 8.5” x 11” flat stitched, some diagrams. Theory with examples for a cube, tesseract and 5-D hypercube.

Hendricks, John R., A Bimagic Cube of Order 25, Self-published, 2000, 0-9684700-7-6. 14 pages plus covers, 8.5” x 11” flat stitched, some diagrams. Coordinate equations and a basic program to generate this cube. A companion booklet by Holger Danielsson shows the horizontal planes of this cube.


Hendricks, John R., Bi-Magic Squares of Order 9, Self-published, 1999, 0-9684700-6-8 14pp + covers, 8.5” x 11”. A method of generating these squares using equations and coefficient matrices.

Kelsey, Kenneth, The Cunning Caliph, Frederick Muller, 1979, 0-584-10367-0 This is one of the five books (the first one) that make up The Ultimate Book of Number Puzzles.

Kelsey, Kenneth The Ultimate Book of Number Puzzles, Cresset Press, 1992, 0-88029-920-7 This is a combination of 5 books ( four by K. Kelsey & the last one by D. King), all published in Great Britain 1979-1984 by Frederick Muller Ltd. It consists of numerical puzzles in the form of magic squares, stars, etc. No theory, but lots of examples (some quite original) and lots of practice material.

Moran, Jim, The Wonders of Magic Squares, Vantage Books, 1982, 0-394-74798-4 A large format book that is simply written with little theory, but demonstrates a large variety of ways to compose magic squares. Contains a forward by Martin Gardner.

Ollerenshaw, K. and Brée, D., Most-Perfect Pandiagonal Magic Squares, Cambridge Univ. Press, 1998, 0-905091-06-X The methods of construction and enumeration of these doubly-even magic squares.

Pickover, Clifford A., The Zen of Magic Squares, Circles, and Stars, Princeton Univ. Press, 2002, 0-691-07041 Completely devoted to magic objects with lots of diagrams.

Swetz, Frank J., Legacy of the Luo Shu, Open Court Publ. 2002, 0-8126-9448-1 All about the order 3 magic square.


The following books have chapters or sections dealing with magic squares (and related subjects).

Ahl, David H., Computers in Mathematics, Creative Computing Pr., 1979, 0-916688-16-X Contains some theory and Basic language programs to generate magic squares. Pages 111-117

Berlekamp, E., Conway, J. and Guy, R., Winning Ways vol. II, Academic Press, 1982, 01-12-091102-7 Original material on order-4 magic squares. Also shows a tesseract with magic vertices. Pages 778-783.

Dudeney, H.E., Amusements in Mathematics, Dover Publ., 1958, 0-486-20473-1 Originally published in 1917. Order 4 classes, Subtraction, multiplication, division, domino, etc. List of first prime # magic squares, etc. Pages 119-27 and 245-247

Falkener, Edward, Games Ancient and Oriental and How to Play Them, Dover Publ., 1961, 0-486-20739-0 First published by Longmans, Green & Co. in 1892, this book contains the original text with no changes, except for corrections. A comprehensive discussion of magic squares circa 100+ years ago. Pages 267-356

Gardner, Martin, 2nd Scientific American Book of Mathematical Puzzles and Diversions ,Simon and Schuster, 1961, 61-12845. Diabolic hypercube (tesseract), diabolic donut, some history, pages 130-140

Gardner, Martin, Incredible Dr. Matrix, Scribners, 1967, 0-684-14669-X Anti-magic, multiplication & division, pyramid, etc. Pages 21, 47, 211,246


Gardner, Martin, Mathematical Carnival, Alfred Knopf, 1975, 0-394-49406-7 Hypercubes, pages 41-54. Magic Stars, pages 55-65

Gardner, Martin Mathematical Puzzles & Diversions, Simon & Schuster, 1959, 59-9501 Chapter 2, Magic With a Matrix, pages 15-22.

Gardner, Martin, New Mathematical Diversions, Simon and Schuster, 1966, 671-20913-2 Euler's spoilers- order-10 Graeco- Latin squares, order-4 playing card magic square. Pages 162-172

Gardner, Martin, Penrose Tiles to Trapdoor Ciphers, Freeman, 1989, 0-7167-1986-X Alphamagic, smith numbers, 3x3 properties, pages 293-305

Gardner, Martin, Scientific American Book of Mathematical Puzzles and Diversions,, Simon and Schuster, 1959, 59-9501. Using magic squares for magic tricks, pages 15-22.

Gardner, Martin, Sixth book of Mathematical Games, Charles Scribner's Sons, 1963, 0-684-14245-7 Magic hexagons, pages 23-25. Consecutive prime s. (using #1) pages 86-87

Gardner, Martin Time Travel & Other Mathematical Bewilderments, Freeman Publ., 1988, 0-7167-1924-X First published enumeration of Order-5 magic squares and information about order-8 magic cubes. Note that Gardner refers to perfect magic cubes. These are what Hendrick's now calls Diagonal magic cubes. Magic Squares & Cubes.

Pps 213-226.


Page A1-7

Heath, Royal Vale, Mathemagic, Dover Publ., 1953 The author copywrited this material in 1933. Some unusual patterns. Pages 87-123.

Hunter, J. & Madachy, J., Mathematical Diversions, Van Nostrand, 1963, Theory of magic squares includes a simple method to produce bimagic squares. Pages 23-34.

Kraitchik, Maurice, Mathematical Recreations, Dover Publ., 1953, 53-9354 (origin publisher. W. W. Norton, 1942) Construction methods, multi-magic, Graeco-Latin, border, order-4 theory, etc. Pages 142-192

Madachy, Joseph S., Mathematics on Vacation, Thomas Nelson Ltd., 1968, 17-147099-0 A good discussion of magic, anti-magic, heterosquare, talisman, etc squares, pages 85-113. Madachy’s Mathematical Recreations, Dover Publ., 1979, 0-486-23762-1 is a page-for-page copy.

Meyer, Jerome S., Fun With Mathematics, World Publ., 1952, 52-8434 A good discussion of bi-grades and upside-down magic squares of order-4. Pages 47 to 54.

Olivastro, Dominic, Ancient Puzzles, Bantam Books, 1993, 0-553-37297-1 On a Turtle Shell, pages 103-125, discuss the Lo Shu ,Pandiagonal, Franklin and composite magic squares. Also magic graphs. However, he erroneously states that no one has yet discovered a magic tesseract.


Pickover, Clifford A., The Wonder of Numbers, Oxford Univ. Press, 2001, 0-19-513342-0, pp 233-239 plus others. Pickover’s usual great stuff!

Rouse Ball, W. & Coxeter, H., Mathematical Recreations & Essays, 12th Edition, Univ. of Toronto Pr., 1974, 0-8020-6138-9. Pages 189-221. This classic work was originally published in 1892. H. S. M. Coxeter brought it up to date with the 1938 publication of the 11th edition, the 12th edition in 1974 and edition 13 in 1987. Chapter 7 is on magic squares.

Rouse Ball, W. & Coxeter, H., Mathematical Recreations & Essays, 13th Edition, Univ. of Toronto Pr., 1987, 0-486-25357-0. Pages 193-221. See above re edition 12.

Stein, Sherman K. Mathematics: The Man-made Universe, 1963, W. H. Freeman, 63-7786 Chap. 12, Orthogonal Tables. Discussion of Graeco-Latin squares and magic squares. Pages 155-174

Spencer, Donald D. Game Playing with Computers, Hayden, 1968, 0-8104-5103-4 Computer programs and magic square theory. Pages 23-107. Card, division, upside down, composite, prime, subtracting, etc. Pages 209-224.

Spencer, Donald D. Game Playing with Basic, Hayden, 1977, 0-8104-5109-3 Computer programs and magic square theory. Pages 119-141.

Spencer, Donald D. Exploring Number Theory With Microcomputers, Camelot, 1989, 0-89218-249-0 Computer programs and magic square theory, geometric, talisman, multiplying, heterosquares, prime, etc. Pages 155-180


Weisstein, Eric W., Concise Encyclopedia of Mathematics, CRC Press, 1999, 0-8493-9640-9 A general mathematical encyclopedia containing more then 14,000 entries so has many on magic square related subjects.

Weisstein, Eric W., Concise Encyclopedia of Mathematics CD-ROM, CRC Press, 1999, 0-8493-1945-5 Contains all of the material in the book, plus interactive graphics and both internal and external hyperlinks.

Games & Puzzles for Elementary and Middle School Mathematics Readings from the Arithmetic Teacher Published by National Council of Teachers of Mathematics, 1975, 0-87353-054-3.

Readings for Enrichment in Secondary School Mathematics, Bordered Magic Squares

Published by National Council of Teachers of Mathematics, 1988, 0-87353-252-X,. Pages 195-199.

Marián Trenkler, Magic Cubes, The Mathematical Gazette, 82, (March, 1998), 56-61.

Marián Trenkler Magic rectangles, The Mathematical Gazette, 83, 2000, 102-105

Marián Trenkler, A construction of Magic Cubes, The Mathematical Gazette, 84, (March, 2000), 36-41.

Marián Trenkler, Magic p-dimensional Cubes of order n not ≡ 2 (mod 4), Acta Arithmetica (Poland) Acta Arithmetica, 92(2000), 189-194.


Page A1-10

The following Articles on Magic Squares appear in Recreational Mathematics Magazine I show these as Title, Author, RMM:issue #:date: pages(s) More Strictly for Squares Miscellaneous authors RMM: # 5 Oct. 1961 p24-29 Add Multiply Magic Squares Walter W. Horner RMM: # 5 Oct. 1961 p30-32 How to Make a Magic Tesserack Maxey Brooke RMM: # 5 Oct. 1961 p40-44 More Strictly for Squares Miscellaneous authors RMM: # 7 Feb. 1962 p14-19 Anti-magic squares J. A. Lindon RMM: # 7 Feb. 1962 p14-19 Geometric magic squares Boris Kordemskii RMM:#13 Feb. 1963 p3-6

The following Articles on Magic Squares appear in Journal of Recreational Mathematics I show these as Title, Author, JRM:volume #:issue #:date:pages(s) Magic Designs Robert B. Ely III JRM:1:1:1968:3-17 A Magic Square William J. Mannke JRM:1:3:1968:139 The Construction of knight Tours T. H. Willcocks JRM 1:4:1968:225-233 Mannke’s Order-8 Square Leigh Janes JRM:2:2:1969:96 Construction of Odd Order Diabolic Magic Squares J.A.H.Hunter JRM:2:3:1969:175-177 Sums of Third-order Anti-magic Squares Charles W. Trigg JRM:2:4:1969:250-254 Triangles With Balanced Perimeters Charles W. Trigg JRM:3:4:1970:255-256 Fifth Order Concentric Magic Squares Charles W. Trigg JRM:4:1:1971:42-44 Doubly Magic Square with Remarkable Subsidiaries Charles W. Trigg JRM:4:3:1971:171-174 Edge Magic and Edge Anti-magic Tetrahedrons Charles W. Trigg JRM:4:4:1971:253-259 Normal Magic Triangles of order-n Terrel Trotter JRM:5:1:1972:28-32 Edge Anti-magic Tetrahedrons With Rotating Triads Charles W. Trigg JRM:5:1:1972:40-42 The Third Order Magic Square Complete John R. Hendricks JRM:5:1:1972:43-50 The Pan-3-agonal Magic Cube John R. Hendricks JRM:5:1:1972:51-52


Page A1-11 Perfectly Odd Squares Monk A. Ricci JRM:5:2:1972:138-142 Latin Squares Under Restrictions and a Jumboization N. T. Gridgeman JRM:5:3:1972:198-202 Magic Squares with Nonagonal & Decagonal Elements Charles W. Trigg JRM:5:3:1972:203-204 The Pan-3-agonal Magic Cube of Order-5 John R. Hendricks JRM:5:3:1972:205-206 Anti-magic Squares With Sums in Arithmetic Progression Charles W. Trigg JRM:5:4:1972:278-280 Graeco-Latin cubes P. D. Warrington JRM:6:1:1973:47-53 Species of Third-Order Magic Squares & Cubes John R. Hendricks JRM:6:3:1973:190-192 Magic Tesseracts & n-dimensional Magic Hypercubes John R. Hendricks JRM:6:3:1973:193-201 Magic Cubes of Odd Order John R. Hendricks JRM:6:4:1973:268-272 Trimagic Squares William H. Benson JRM:7:1:1974:8-13 Perimeter Magic Polygons Terrel Trotter Jr. JRM:7:1:1974:14-20 Third Order Square Related to Magic Squares Charles W. Trigg JRM:7:1:1974:21-22 Eight Digits on a Cubes Vertices Charles W. Trigg JRM:7:1:1974:49-55 Pan-n-agonals in Hypercubes John R. Hendricks JRM:7:2:1974:95-96 Not Every Magic Square is a Latin Square Joseph M. Moser JRM:7:2:1974:97-99 Some Properties of Third Order Magic Squares Charles W. Trigg JRM:7:2: 1974:100-101 9-digit Determinants equal to Their 1st Rows Charles W. Trigg JRM:7:2: 1974:136-139 A Pandiagonal Magic Square of Order-8 John R. Hendricks JRM:7:3:1974:186 Magic Square Time John R. Hendricks JRM:7:3:1974:187-188 Perfect Magic Cubes of Order Seven Bayard E. Wynne JRM:8:4:1975:285-293 Infinite Magic Squares Ronald J. Lanaster JRM:9:2:1976:86-93 58. Magic Squares Rudolf Ondrejka JRM:9:2:1976:128-129 Perfect Magic Icosapentacles Baynard E. Wayne JRM:9:2:1976:241-248 Pan-diagonal Associative Magic Cubes… Ian P. Howard JRM:9:4:1976:276-278 Related Magic Squares with Prime Elements Gakuho Abe JRM:10:2:1977:96-97


Computer Constructed Magic Cubes Ronald J. Lancaster JRM:10:3:1977:202-203 Magic Talisman Squares Greg Fitzgibbon JRM:10:4:1977:279-280 Perimeter Antimagic Tetrahedrons Charles W. Trigg JRM:11:2:1978-79:105- Magic Cubes of Prime Order K.W.H.Leeflang JRM:11:4:1978-79:241-257 The Perfect Magic Cube of Order-4 John R. Hendricks JRM:13:3:1980-81:204-206 A Family of Sixteenth Order Magic Squares . Charles W. Trigg JRM:13:4:1980-81:269-273 The Pan-3-agonal Magic Cube of Order-4 John R. Hendricks JRM:13:4:1980-81:274-281 Consecutive-Prime Magic Squares Alan W. Johnson Jr. JRM:14:2:1981-82:152-153 Special Anti-magic Triangular Arrays Charles W. Trigg JRM:14:4:1981-82:274-278 Consecutive-Prime Magic Squares Alan W. Johnson Jr. JRM:15:1:1982-83:17-18 A Bordered Prime Magic Square Alan W. Johnson Jr. JRM:15:2:1982-83:84 The Construction of Doubly-even Magic Squares Tien Tao Kuo JRM:15:2:1982-83:94-104 A Unique 9-Digit Square Array Vittorio Fabbri JRM:15:3:1982-83:170-171 A Sixth Order Prime Magic Square Alan W. Johnson Jr. JRM:15:3:1982-83:199 Magic Triangular Regions of Orders 5 and 6 Katagiri & Kobayashi JRM:15:3:1982-83:200-208 Irregular Perfect Magic Squares of Order 7 Gakuho Abe JRM:15:4:1982-83:249-250

Early Books on Magic Squares William L. Schaaf JRM:16:1:1983-84:1-6

666 – Order 4 M. S. (Letter to the Editor) Rudolf Ondrejka JRM:16:2:1983-84:121 9-digit Digit-Root Magic & Semi-Magic Squares Charles W. Trigg JRM:17:2:1985:112-118 Ten Magic Tesseracts of Order Three John R. Hendricks JRM:18:2:1986:187-188 A Magic Rooks Tour Stanley Rabinowitz JRM:18:3:1986:203-204 Letter to the Editor Borders for 2nd-order square John R. Hendricks JRM:19:1:1987:42 Generating a pandiagonal Magic Square of Order-8 John R. Hendricks JRM:19:1:1987:55-58 Vestpocket Biblio. No. 12. Magic Squares and Cubes William L. Schaaf JRM:19:2:1987:81-86 A Ninth Order Magic Cube John R. Hendricks JRM:19:2:1987:126-131


Page A1-13

Constructing Pandiagonal Magic Squares of Odd Order John R. Hendricks JRM:19:3:1987:204-208 Creating Pan-3-agonal Magic Cubes of Odd Order John R. Hendricks JRM:19:4:1987:280-285 Some Ordinary Magic Cubes of Order 5 John R. Hendricks JRM:20:1:1988:125-134 A Magic Cube of Order 7 John R. Hendricks JRM:20:1:1988:23-25 Related Magic Squires Alan W. Johnson Jr. JRM:20:1:1988:26 Pandiagonal Magic Squares of Odd Order John R. Hendricks JRM:20:2:1988:81-86 Magic Cubes of Odd Order by Pocket Computer John R. Hendricks JRM:20:2:1988:87-91 The Diagonal Rule for Magic Cubes of Odd Order John R. Hendricks JRM:20:3:1988:192-195 More Pandiagonal Magic Squares John R. Hendricks JRM:20:3:1988:198-201 A Consecutive Prime 3 x 3 Magic Square Harry L. Nelson JRM:20:3:1988:214-216 The Third Order Magic Tesseract John R. Hendricks JRM:20:4:1988:251-256 Another Magic Tesseract of Order-3 John R. Hendricks JRM:20:4:1988:275-276 Creating More Magic Tesseracts of Order-3 John R. Hendricks JRM:20:4:1988:279-283 Groups of Magic Tesseracts John R. Hendricks JRM:21:1:1989:13-18 More and More Magic Tesseracts John R. Hendricks JRM:21:1:1989:26-28 The Pan-4-agonal Magic Tesseract of Order-4 John R. Hendricks JRM:21:1:1989:56-60 A Perfect 4_Dimensional Hypercube of Order-7 Arkin Arney & Porter JRM:21:2:1989:81-88 Palindromes and Magic Squares Alan W. Johnson Jr. JRM:21:2:1989:97-100 Supermagic and Antimagic Graphs N.Hartsfield & G. Ringel JRM:21:2:1989:107-115 The Determinant of a Pandiagonal Magic Square is 0 John R. Hendricks JRM:21:3:1989:179-181 A 5-Dimensional Magic Hypercube of Order-5 John R. Hendricks JRM:21:4:1989: 245-248 The Magic Tesseracts of Order-3 Complete John R. Hendricks JRM:22:1:1990: 15-26 The Secret of Franklin’s 8 x 8 Magic’ Square Lalbhai D. Patel JRM:23:3:1991:175-182 Magic Squares Matrices Planes and Angles Frank E. Hruska JRM:23:3:1991:183-189 Minimum Prime Order-6 Magic Squares Alan W. Johnson Jr. JRM:23:3:1991:190-191


Inlaid Odd Order Magic Squares John R. Hendricks JRM:24:1:1992:6-11 A Note on Magic Tetrahedrons John R. Hendricks JRM:24:4:1992:244 Prime Magic Squares for the Prime Year 1993 Alan W. Johnson Jr. JRM:25:2:1993:136-137 An Inlaid Magic Cube John R. Hendricks JRM:25:4:1993:286-288 Property of Some Pan-3-Agonal Magic Cubes of Odd Order John R. Hendricks JRM:26:2:1994:96-101 Inlaid Pandiagonal Magic Squares John R. Hendricks JRM:27:2:1995:123-124 Inlaid Magic Squares John R. Hendricks JRM:27:3:1995:175-178 More Magic Squares Emanuel Emanouilidis JRM:27:3:1995:179-180 More Multiplication Magic Squares Emanuel Emanouilidis JRM:27:3:1995:181-182 Powers of Magic Squares Emanuel Emanouilidis JRM:29:3:1998:176-177 Palindromic Magic Squares Emanuel Emanouilidis JRM:29:3:1998:177-178 Note on the Bimagic Square of Order-3 John R. Hendricks JRM:29:4:1998:265-267 Magic Squares of Order-4 and Their Magic Square Loops Robert S. Sery JRM:29:4:1998:265-267 A Partial Magic Tesseract of Order Two John R. Hendricks JRM:29:4:1998:290-291 Magic Reciprocals Jeffrey Haleen JRM:30:1:1999:72-73 Magic Diamond for the New Millenium E. W. Shineman, Jr. JRM:30:2:1999:112 From Inlaid Squares to Ornate Cube John R. Hendricks JRM:30:2:1999:125-136 Smarandache Magic Problem 2466 sloution Charles Ashbacher JRM:30:4:1999:296-299 A Purely Pandiagonal 4x4 Square and the Myers-Briggs … Peter D. Loly JRM:31:1:2002-2003:29-31 Concatenation on Magic Squares Emanuel Emanouilidis JRM:31:2:2002:110-111 Franklin’s “Other” 8-Square Paul C. Pasles JRM:31:3:2002:161-166 2617 Solution: Magic Cube of Primes Harvey d. Heinz JRM:31:4:2002: 298 A Unified Classification System for Magic Hypercubes H. Heinz and J. Hendricks JRM:32:1:2003-2004: 30-36 Square-Ringed Magic Squares Jeffrey Heleen JRM:32:2:2003:144-146 k-Sets of nth Order Magic Squares D. Fell and A. Shulman JRM:32:3:2003:181-192


Page A1-15

Literature On Magic Stars In contrast to the voluminous literature for magic squares spanning 100's of years, there has been very little published on magic stars. The main sources of information I have been able to locate are: H.E.Dudeney, 536 Puzzles & Curious Problems, Scribner's 1967. Pages 145-147 and 347-352. Martin Gardner, Mathematical Recreations column of Scientific American, Dec. 1965, reprinted with addendum in Martin

Gardner, Mathematical Carnival, Alfred A. Knoff, 1975. Mostly on order 6, but mention made of total basic solutions for orders 7 & 8 (also corrected number for order-6). Marián Trenkler of Safarik University, Kosice, Slovakia published a paper on Magic Stars. It is called "Magicke hviezdy" (Magic stars) and appeared in Obsory matematiky, fyziky a informatiky, 51(1998), pages 1-7. (Obsory = horizons (or line of sight) of mathematics, physics and informatics. Perfect Magic Icosapentacles Bayard E. Wynne JRM:9:4:1976:241-248 Anti-magic Pentagrams Charles W. Trigg JRM 10:3::1977:169-173 A Magic Asteriod Gakuho Abe JRM:16:2:1983-84:113 Magic Pentagram Solutions Over GF(2) Harold Reiter JRM:20:2:1988:99-104 Two New Magic Asteroids Laurent Hodges JRM:24:2:1992:85-86 The Magic Hexagram John R. Hendricks JRM:25:1:1993:10-12 Letter to the Editor (Magic Asteroids) Alan W. Johnson Jr. JRM:26:2:1994:90-91 Almost Magic Charles W. Trigg JRM:29:1:1998:8-11 The Heinz Web site has 17 pages (currently) on magic stars. They start at http://www.geocities.com/~harveyh/magicstar.htm

The ten nodes of this graph are each connected to all the other nodes with lines (edges) labeled with the consecutive integers from 1 to 45. All 9 lines connected to each node sum to the same magic constant.

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Appendix 2-1

Bibliography of

Articles written by John R. Hendricks (Not included – material in Appendix 1)

STATISTICAL ARTICLES

1. Extreme Temperature Recurrence, Atmospheric Environment Service, [AES] TEC 801, 15 February 1974.

2. Probability and Time, Statistical Association of Manitoba, [SAM] Newsletter Vol.3, No. 3, January 1980.

3. Notes on Probability and Time, SAM Newsletter, Vol. 3, No. 4, March 1980.

4. Probability Mean Time, AES, Central Region, Technical Notes, No. 83-1, January 17th 1983.

5. The Standardized Normal Distribution Function and Your Pocket Computer, The Manitoba Mathematics Teacher, Vol. 15, No. 4, April 1987.

6. Application of Cube-Root, The Manitoba Mathematics Teacher, Vol. 16, No. 1, September 1987.

7. The Statistical Probability of Temperatures and Your Pocket Computer, SAM newsletter Vol.11, No. 5, 11 January 1988.

8. The Statistical Probability of Precipitation and Your Pocket Computer, SAAAM Newsletter, Vol. 11, No. 6, 1 March 1988.

9. Probability Mean Time, The Manitoba Mathematics Teacher, Vol. 16, No. 3, March 1988.

10. Probability and Time, Self-published handout material for a talk to the joint meeting of the Statistical Association of Manitoba and the American Statistical Society, Red River Chapter held on the 16th day of April 1988 at the Senate chambers of the University of Manitoba,

11. The Statistical Analysis of the Circumference of an Ellipse, SAM Newsletter, Vol. 14 No.5, 14th January 1991.

12. Elliptic Arc Lengths, SAM Newsletter, Vol. 14, No. 7, 18 April 1981.

13. An Analysis of the 1993 Summer Rainfall at Winnipeg, AES, October 1993, Internal Report Number SSD-W93-01, Scientific Services, Central Region.

A2-2 METEOROLOGY

14. Meteorology, Exercises for Students at R.C.A.F. Station Gimli, Single A.F.S., Queens Printer, Winnipeg, 1957, Classified.

15. A Problem in Single-Heading Flight for Jet Aircraft, AES, Circ. 3549, TEC 374, 15 October 1981, pp. 1-19

16. Forecasting Takeoff Pressures, AES Circ. 3777, TEC 444, 19th December 1962.

17. The Diurnal Pressure Graph, AES, Circ. 3802, TEC. 453, 19 February 1963.

18. Using the Normal Curve for Temperature Frequencies, AES, TEC 765, 25 February 1972.

19. Precipitation at Regina in June, AES, TEC 772, 18 July 1972. 20. A Mirage, Zephyr Magazine, December 1972. 21. A Mirage at Regina, Atmosphere, Vol. 1,1No. 1, 1973. 22. A Probability Study of Extreme Temperatures Part I, Theory,

AES Tec 789, 14 August 1973. 23. A Probability Study of Extreme Temperatures Part II, Application

to High Maxima, AES Tec. 790, 14 August 1973 24. A Probability Study of Extreme Temperatures Part III,

Application to Low Minima, High Minima and Low Maxima, AES, TEC 791, 14 August 1973.

25. A Probability Study of Extreme Temperatures Part IV. Further Study, AES, TEC 792, 16 November 1973.

26. A Probability Study of Extreme Temperatures Part V, Results and General Conclusiions, AES, TEC 806, 15 February 1974

27. The Frequency of Thunderstorm Days at Regina, AES, TEC 799, 4 December 1973.

28. Precipitation Probabilities at Regina, AES, TEC 809, 5 Septembeer 1974.

29. Consistency in Forecast Precipitation Probabilities, AES, Central Region Technical Notes, No. 82-2, Nov. 28, 1982.

A2-3 MISCELANEOUS ARTICLES

30. Magic Forms, Magic Squares: VOXAIR, an Airforce Newsmagazine, 4 February .

31. Magic Forms: Magic Cubes, VOXAIR, an Airforce Newsmagazine, 14 May 1955.

32. Russian Peasant Multiplication, VOXAIR, 23 August 1955 33. The Canadian Monarchy, Saskatchewan Genealogical Society

Bulletin, Vol. 9, No. 1, 1978. 34. The Soul, The Ark, Vol. XLVIII, No. 1, (No. 129), April 1980. 35. The Animal, The Ark, Vol. XLVIX, No. 1, (No. 132), Apr. 1981. 36. Animal Group Names, The Ark, Vol. XLVIX, No. 1, (No. 132) ,

April 1981. 37. A Straight Line, Journal of Recreational Mathematics, Vol. 14(2),

1982 38. The High School Mathematics Club: A Model. The Manitoba

Mathematiccs Teacher, Vol. 14, No. 4, April 1986. 39. The Third-Order Magic Square Complete, The Manitoba

Mathematics Teacher, Vol. 15, No. 2, December 1986. 40. Large Factorial, Journal of Recreational Mathematics, Vol. 21(2),

1989 41. Large Factorial, The Manitoba Mathematics Teacher, Vol. 16,

No. 3, March 1988. 42. Pythagoras Theorem, The Manitoba Mathematics Teacher, Vol.

17, No. 2, January 1989. 43. Magic Cube Terminology, The Manitoba Mathematics Teacher,

Vol. 17, No. 2, January 1989. 44. A Point to Remember, The Manitoba Mathematics Teacher, Vol.

17, No. 3, March 1989. 45. Inlaying Magic Squares, The Manitoba Mathematics Teacher,

Vol 17. No. 4, May 1989. 46. Correcting Imbalance, The Ark, No. 172, Winter 1994..

NOTE (2005): John Hendricks books shown on the following pages are now all out-of-print. NOTE (2006): The Magic Square Lexicon: Illustrated Is still available (2nd run with corrections). Order via email in Advertisement or online at http://www.geocities.com/~harveyh/

Magic Squares to Tesseracts by Computer

by John R. Hendricks ISBN 09684700-0-9, 31 December 1998

$32.00 CAN each, or $25.00 U.S. each 206 pages run on 22x28 cm paper, spiral bound,

card covers, indexed.

This book shows the method of construction of magic squares, magic cubes, and magic tesseracts by the use of modular equations. Matrices are in the appendix for those people wishing to do it that way. There is a large section on geometry. The BASIC program is given, that the author used in order to achieve the results in his TI –74 BASICALC programmable battery operated calculator. You might have to adapt the programs with slight changes to your particular computer. Many pages are devoted to showing the various aspects of cubes and tesseracts because of rotations and reflections.

Shown and explained are the following:

•What is meant by a pandiagonal magic cube.

•What is meant by a pantriagonal magic cube

•What is meant by a PERFECT cube.

•Magic tesseracts of orders 3, 4, 5 are shown

•The perfect magic tesseract of order 16 is explained in a separate paper, but the ground work is prepared..

Books are sold privately by John R. Hendricks to those people who are interested. He may be contacted by e-mail at [email protected] or by phone at (250)-381-1544, or through his address:

John R. Hendricks #308 – 151 St. Andrews St. Victoria, B.C., V8V 2M9,

CANADA

Inlaid Magic Squares and Cubes, 2nd Edition

by John R. Hendricks

Edited by: Holger Danielsson

ISBN 0-9684700-3-3, 14 July 2000.

$40.00 CAN each, or $32.00 U.S. each

255 pages on 22x28 cm paper, spiral bound, card covers, indexed.

This book is especially written for the young people in Junior and Senior high schools who are interested in mathematics. It is meant as a source book or REFERENCE book. The method brings in algebraic digits, rather than algebraic numbers.. Most of the innovations are due to the author’s very own lifetime pursuit of bigger and better magic squares.

But, even if you have not the time to actually make them yourself, where else can you obtain such a wide variety of almost every conceivable type of magic square.

As you know, magic squares of orders 6, 10, 14, etc. are much more difficult to make than other orders. Well, there is a new technique for making them in the appendix of the book. There are:

• Magic squares with inlaid diamonds

• The world’s first inlaid magic cubes

• The first magic square with interchangeable parts

• Bimagic squares & inlaid bimagic squares

• Magic squares of double order

• Patchwork squares

Books are sold privately by John R. Hendricks to those people who are interested. He may be contacted by e-mail at

[email protected], by phone at (250)-381-1544

Various Small Booklets for Sale by John R. Hendricks

$8.00 CAN each, or $6.00 U.S. each

36 pages run on 22x28 cm paper, stapled, card covers

All Third Order Magic Tesseracts ISBN 0-9694700-2-5, 15 February 1999

This booklet contains the 58 basic magic tesseracts as well as the four basic magic cubes of order 3.

Perfect n-Dimensional Magic Hypercubes

of Order 2n

ISBN 0-9684700-4-1, 21 May 1999

This booklet contains the theory and the BASIC programs for the TI74 BASICALC calculator to produce both the 16th order magic tesseract, and the 32nd order 5-dimensional magic hypercube, as well. A complete review is made of the definitions of “perfect” and why the new definition must prevail.

Curves and Approximations ISBN 0-9684700-5-X, 4 September 1999

This booklet contains sundry papers, (non-magic square ones,) which the author has published elsewhere

. Many curves cannot be plotted by the curve-plotting calculators that abound. This is because x cannot be solved for y, or y for x. There is another way. Teachers teach the area of an ellipse, but not the circumference, even though a very close approximation is available. There are many new curves which are shown that have loops on them, such as the Cosine Nodosus. Booklets are sold privately by John R. Hendricks to those people who are interested. He may be contacted by e-mail at magic-

[email protected], or by phone at (2250)-381-1544, or through his address:

A Bimagic Cube: Order 25

by John R. Hendricks ISBN 0-9684700-7-6, 9 June 2000

$5.00 CAN each, or $4.00 U.S. each 14 pages run on 22x28 cm paper, stapled, card covers.

This is the world’s first Bimagic Cube . It is of order 25, which means that it contains 253 numbers. That is 15,625 numbers. The magic sum is 195,325. But, that is not all. If you square all the numbers in the cube and then add it up, the sum of the squares is a constant 2,034,700,525. Now, we are left wondering if there is a smaller bimagic cube. This booklet provides the theory and the program in BASIC for the TI-74. BASICALC calculator.

Printout Of a Bimagic Cube: Order 25 by Holger Danielsson

$8.00 CAN each, or $6.00 U.S. each 36 pages run on 22x28 cm paper, stapled, card covers

However, Holger Danielsson , a teacher of mathematics and computer science in a high school in Germany, kindly provided a layer-by-layer printout of the cube, including each face and more explanation of the geometry involved. Some people will likely want the theory. Some may like only the cube. Others may wish both. Therefore, there is a deal:

Special Deal: One of each

$12.00 CAN, or $9.00 U.S. The booklets really go together as a pair.

Books are sold privately by John R. Hendricks to those people who are interested. He may be contacted by e-mail at [email protected], by phone at

(250)-381-1544

ORDER THIS BOOK

Magic Square Lexicon: Illustrated

written by H.D. Heinz and John R. Hendricks

is a true reference book. It has been put together by two men both of whom have a lifetime of knowledge and experience in magic squares, cubes and related items. Harvey Heinz is a mathematical hobbyist. He has pursued magic circles, spheres, stars, polygrams and a wide range of

other mathematical novelties and oddities. John Hendricks by contrast, decided to pursue magic

hypercubes in higher dimensional spaces and to unravel and publicize their mystery.

Fully explained with the help of diagrams and tables is the new concept of perfect as applied to magic squares, cubes,

tesseracts, etc.

There is nothing else on the

market like it. Everything you wanted to know and more.

Definitions ◊ Tables ◊ Examples

◊ Illustrations ◊ Terminology ◊ Included are two appendices of bibliographies. ISBN 0-9687985-0-0, 228 pages 5 ¼ x 8, Perfect bound.

WRITE FOR YOUR COPY TODAY!

Please remit $32 Canadian, or $25 U.S. funds per copy to:

H..D. Heinz,

15450 92A Avenue

Surrey, BC, V3R 9B1, Canada E-mail: [email protected]

This book defines 239 terms associated with magic squares, cubes,tesseracts, stars, etc. Many of these terms have been in use hundredsof years while some were coined in the last several years. Whilemeant as a reference book, it should be ideal for casual browsing, withits almost 200 illustrations and tables, 171 of which are captioned.

While this book is not meant as a "how-to do" book, it should be asource of inspiration for anyone interested in this fascinating subject.Many tables compare characteristics between orders or dimensions.The illustrations were chosen, where possible, to demonstrateadditional features besides the particular definition.

$ 32.00 Cdn. $ 25.00 U.S.ISBN 0-9687985-0-0

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