Table of Contents

Foreword ............................................................................................................... 9

1. Introduction ...................................................................................................... 171.1 The general Constraint Satisfaction Problem (CSP)................................... 171.2 Paradigms of resolution .............................................................................. 201.3 Parameters and instances of a CSP; minimal instances; classification....... 241.4 The basic and the more complex resolution theories of a CSP................... 261.5 The roles of Logic, AI, Sudoku and other examples .................................. 281.6 Notations ..................................................................................................... 34

PART ONE: LOGICAL FOUNDATIONS........................................................ 35

2. The role of modelling, illustrated with Sudoku ............................................. 372.1 Symmetries, analogies and supersymmetries ............................................. 372.2 Introducing the four 2D spaces: rc, rn, cn and bn....................................... 422.3 CSP-Variables associated with the rc, rn, cn and bn spaces....................... 482.4 Introducing the 3D nrc-space...................................................................... 49

3. The logical formalisation of a CSP.................................................................. 513.1 A quick introduction to Multi-Sorted First Order Logic (MS-FOL) .......... 513.2 The formalisation of a CSP in MS-FOL: T(CSP)....................................... 583.3 Remarks on the existence and uniqueness of a solution ............................. 633.4 Operationalizing the axioms of a CSP Theory ........................................... 643.5 Example: Sudoku Theory, T(Sudoku) or ST .............................................. 653.6 Formalising the Sudoku symmetries........................................................... 703.7 Formal relationship between Sudoku and Latin Squares............................ 73

4. CSP Resolution Theories.................................................................................. 754.1 CSP Theory vs CSP Resolution Theories; resolution rules ........................ 764.2 The logical nature of CSP Resolution Theories.......................................... 774.3 The Basic Resolution Theory of a CSP: BRT(CSP) .................................. 864.4 Formalising the general concept of a Resolution Theory of a CSP............ 884.5 The confluence property of resolution theories .......................................... 894.6 Example: the Basic Sudoku Resolution Theory (BSRT)............................ 914.7 Sudoku symmetries and the three fundamental meta-theorems ................. 94

6 Pattern-Based Constraint Satisfaction and Logic Puzzles

PART TWO: GENERAL CHAIN RULES........................................................ 99

5. Bivalue chains, whips and braids ....................................................................1015.1 Bivalue chains.............................................................................................1025.2 z-chains, t-whips and zt-whips (or whips) ..................................................1035.3 Braids ..........................................................................................................1085.4 Whip and braid resolution theories; the W and B ratings...........................1105.5 Confluence of the Bn resolution theories; resolution strategies ..................1125.6 The “T&E vs braids” theorem ....................................................................1155.7 The objective properties of chains and braids.............................................1195.8 About loops in bivalue-chains, whips and braids .......................................1245.9 Forcing whips, a bad idea?..........................................................................1265.10 Exceptional Sudoku examples ..................................................................1275.11 Whips in N-Queens and Latin Square; definition of SudoQueens ..........144

6. Unbiased statistics and whip classification results ........................................1536.1 Classical top-down and bottom-up generators............................................1556.2 A controlled-bias generator.........................................................................1566.3 The real distribution of clues and the number of minimal puzzles.............1616.4 The W-rating distribution as a function of the generator............................1636.5 Stability of the classification results ..........................................................1646.6 The W rating is a good approximation of the B rating ...............................165

7. g-labels, g-candidates, g-whips and g-braids..................................................1677.1 g-labels, g-links, g-candidates and whips[1]...............................................1677.2 g-bivalue chains, g-whips and g-braids ......................................................1717.3 g-whip and g-braid resolution theories; the gW and gB ratings .................1747.4 Comparison of the ratings based on whips, braids, g-whips and g-braids..1767.5 The confluence property of the gBn resolution theories .............................1797.6 The “gT&E vs g-braids” theorem ...............................................................1837.7 Exceptional Sudoku examples ....................................................................1847.8 g-labels and g-whips in N-Queens and in SudoQueens..............................1977.9 g2-whips and g2-braids...............................................................................201

PART THREE: BEYOND G-WHIPS AND G-BRAIDS..................................203

8. Subset Rules in a general CSP.........................................................................2058.1 Transversality, Sp-labels and Sp-links .........................................................2068.2 Pairs.............................................................................................................2088.3 Triplets ........................................................................................................2118.4 Quads ..........................................................................................................2148.5 Relations between Naked, Hidden and Super Hidden Subsets in Sudoku .2208.6 Subset resolution theories in a general CSP; confluence............................222

Table of Contents 7

8.7 Whip subsumption results for Subset rules.................................................2248.8 Subsumption and non-subsumption examples from Sudoku......................2268.9 Subsets in N-Queens ...................................................................................236

9. Reversible-Sp-chains, Sp-whips and Sp-braids................................................2399.1 Sp-links; Sp-subsets modulo other Subsets; Sp-regular sequences ..............2409.2 Reversible-Sp-chains ...................................................................................2439.3 Sp-whips and Sp-braids................................................................................2489.4 The confluence property of the SpBn resolution theories............................2559.5 The “T&E(Sp) vs Sp-braids” theorem, 1≤p≤ ............................................2599.6 The scope of Sp-braids (in Sudoku) ............................................................2619.7 Examples.....................................................................................................263

10. g-Subsets, Reversible-gSp-chains, gSp-whips and gSp-braids .....................26710.1 g-Subsets ...................................................................................................26810.2 Reversible-gSp-chains, gSp-whips and gSp-braids ....................................27710.3 A detailed example ...................................................................................286

11. Wp-whips, Bp-braids and the T&E(2) instances ..........................................29111.1 Wp-labels and Bp-labels; Wp-whips and Bp-braids ...................................29111.2 The confluence property of the BpBn resolution theories .........................30311.3 The “T&E(Bp) vs Bp-braids” and “T&E(2) vs B-braids” theorems..........30811.4 The scope of Bp-braids in Sudoku… ........................................................31211.5 Existence and classification of instances beyond T&E(2)........................318

12. Patterns of proof and associated classifications...........................................32712.1 Bi-whips, bi-braids, confluence and bi-T&E............................................32812.2 W*p-whips and B*p-braids........................................................................33512.3 Patterns of proof and associated classifications........................................34112.4 d-whips, d-braids, W*d-whips and B*d-braids..........................................354

PART FOUR: MATTERS OF MODELLING ..................................................357

13. Application-specific rules (the sk-loop in Sudoku)......................................35913.1. The EasterMonster family of puzzles and the sk-loop.............................36013.2. How to define a resolution rule from a set of examples ..........................36213.3. First interpretation of an sk-loop: crosses and belts of crosses................36313.4. Second interpretation of an sk-loop: x2y2-chains ...................................36813.5. Should the above definitions be generalised further? ..............................37013.6. Measuring the impact of an application-specific rule..............................37313.7. Can an (apparently) application-specific rule be made general? .............374

14. Transitive constraints and Futoshiki ............................................................37514.1 Introducing Futoshiki and modelling it as a CSP .....................................375

8 Pattern-Based Constraint Satisfaction and Logic Puzzles

14.2 Ascending chains and whips.....................................................................37814.3 Hills, valleys and S-whips.........................................................................38314.4 A detailed example using the hill rule, the valley rule and Subsets .........38514.5 g-labels, g-whips and g-braids in Futoshiki ..............................................39114.6 Modelling transitive constraints................................................................39714.7 Hints for further studies on Futoshiki .......................................................398

15. Non-binary arithmetic constraints and Kakuro ..........................................40115.1 Introducing Kakuro...................................................................................40215.2 Modelling Kakuro as a CSP......................................................................40915.3 Elementary Kakuro resolution rules and theories .....................................41615.4 Bivalue-chains, whips and braids in Kakuro ............................................42115.5 Theory of g-labels in Kakuro....................................................................42915.6 Application-specific rules in Kakuro: surface sums .................................43715.7 An exceptional puzzle with very different W and gW ratings..................442

16. Topological and geometric constraints: map colouring and path finding 44516.1 Map colouring and the four-colour problem.............................................44516.2 Path finding: Numbrix® and Hidato®.........................................................449

17. Slitherlink: global constraints and macro-rules ..........................................46717.1 The Slitherlink (or Fences) puzzle: definitions and generalities ..............46717.2 Modelling Slitherlink as a Constraint Satisfaction Problem.....................47117.3 Conventions, rule names and preliminary theorems.................................48517.4 “Classical” resolution rules provable in BRT or W1 – anywhere .............49317.5 “Classical” resolution rules provable in BRT or W1 – grid corners .........50717.6 “Classical” resolution rules provable in BRT or W1 – (pseudo-)edges ....50817.7 “Classical” resolution rules not provable in W1 – anywhere ....................51117.8 “Classical” resolution rules not provable in W1 – grid corners ................51517.9 “Classical” resolution rules not provable in W1 – (pseudo-)edges ...........52017.10 More general rules related to the only-one-loop constraint ....................52017.11 Slitherlink resolution theories, examples and pseudo-statistics..............523

18. Final remarks..................................................................................................53918.1 About our approach to the finite CSP .......................................................53918.2 About minimal instances, uniqueness, density of constraints...................54518.3 About ratings, simplicity, patterns of proof ..............................................54918.4 About CSP-Rules ......................................................................................553

References..............................................................................................................559Books and articles .............................................................................................559Websites............................................................................................................561