The Hidden Logic of Sudoku

"A new conceptual framework for solving Sudoku puzzles with pure logic"

Denis Berthier

book cover

Second, revised and extended edition, November 2007
29.95 euros    (paperback, 416 pages, Royal: 6.14" x 9.21" - 15.6 cm x 23.4 cm)

ISBN : 978-1-84799-214-7 Publishers

Support print-on-demand publishing, buy this book directly from Lulu

Keywords: Artificial Intelligence, Constraint Satisfaction, Constructive Logic, Logic Puzzles, Sudoku.

(First edition published by, May 2007, ISBN : 978-1-84753-472-9)

 © Denis Berthier. All the material in this page and the pages it gives access to are the property of the author and may not be re-published or re-posted without his prior written permission.

     "The Hidden Logic of Sudoku" provides the first systematic perspective of the logical foundations and of the symmetries of the popular game. These are fully exploited to define new kinds of resolution rules, new graphical representations that may ease the solving process and a precedence ordering of the rules consistent with their logical complexity.
     In addition to a few elementary rules, the very classical and basic pattern of xy-chains has been extended, as far as its underlying logic allowed, to a homogeneous set of chain rules of progressively increasing complexity. It suffices to solve almost any puzzle without making guesses, without dealing with chains of subsets or with nets and without assuming the uniqueness of a solution.
     These rules are illustrated with a hundred puzzles, together with their full resolution paths. They have been tested in an Artificial Intelligence (AI) engine and tens of thousands of puzzles have been processed, leading to a precise evaluation of the efficiency of each rule.
     This book is intended for Sudoku players of all levels: they will discover many new facets of the game and new types of resolution rules - introduced in a pedagogical way and set in a uniform conceptual framework based on patterns. It is also intended for teachers or students of Logic or AI: they will appreciate the strict logical foundations.

What's new in the second edition (pdf)

Contents (pdf)

Introduction (pdf)

Conclusion (pdf)

Backcover (pdf)

Press release (pdf)

Press release (french version) (pdf)

Extended Sudoku Board (pdf): This is a Sudoku Board specifically designed for the easy application of the Sudoku resolution method defined in the book. Using this board, complex hidden patterns will be made obvious. How it can be built and used is described in full detail in the book.
This board is copyrighted with the only goal that it will not be patented by anybody and will always remain available to the whole Sudoku community. Feel free to use it and to tell me about your experience with it.

Frequently Asked Questions

<> p. 48, section III.1.6.1, a bug in the Lulu pdf generator makes the sort axioms illegible. They should read:
forall n (n=1n ∨ n=2n ∨ n=3n ∨ n=4n ∨ n=5n ∨ n=6n ∨ n=7n ∨ n=8n ∨ n=9n)
forall r (r=1r ∨ r=2r ∨ r=3r ∨ r=4r ∨ r=5r ∨ r=6r ∨ r=7r ∨ r=8r ∨ r=9r)
forall c (c=1c ∨ c=2c ∨ c=3c ∨ c=4c ∨ c=5c ∨ c=6c ∨ c=7c ∨ c=8c ∨ c=9c)
forall b (b=1b ∨ b=2b ∨ b=3b ∨ b=4b ∨ b=5b ∨ b=6b ∨ b=7b ∨ b=8b ∨ b=9b)
forall s (s=1s ∨ s=2s ∨ s=3s ∨ s=4s ∨ s=5s ∨ s=6s ∨ s=7s ∨ s=8s ∨ s=9s)
forall ut (ut=row ∨ ut=col ∨ ut=blk)

<> p. 275 of HLS1, p. 278 of HLS2, there is a typo in the graphical representations of puzzle Sudogen0-456. Thanks to Gordon Fick for signalling it. Digits 4 and 5 avec been interverted in row 5. The correct puzzle, its L1_0 elaboration and its solution shoud read respectively (where 0 stands for an empty cell):

      904006000          914506000          914526738
      000000501          000900541          263987541
      007040000          007041000          857341629
      002794100          002794100          382794156
      000060090          040860090          741865392
      600100804          600100804          695132874
      000000000          000000000          126473985
      030018000          030018000          539218476
      408059200          408059210          478659213


for advanced players (not the good place to start with)

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