Using Constraint Processing to Model,
Solve, and Support Interactive Solving
of Sudoku Puzzles
Christopher G. Reeson
Constraint Systems Laboratory
Department of Computer Science & Engineering
University of Nebraska-Lincoln
Research supported by a UCARE grant
Constraint Systems Laboratory
March 26, 2007
Reeson–Undergraduate Thesis
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Goals
• How to use CP techniques on Sudoku puzzles?
– Study & compare the effectiveness of various
constraint propagation algorithms
• How can we use CP to interactively support and
guide human players?
• Is Sudoku useful to illustrate and teach CP?
• … How to deconstruct Sudoku solving to
– Demystify the puzzle
– Prevent students from losing time playing it
Constraint Systems Laboratory
March 26, 2007
Reeson–Undergraduate Thesis
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Outline
• Background information: the game, CSPs
• Our contributions
– Algorithms
– Conjectures
– Interface: interactive solver
• Conclusions and future research
• Live demonstration
Constraint Systems Laboratory
March 26, 2007
Reeson–Undergraduate Thesis
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The game
• Number place
• Main properties
– NP-complete
– Well-formed Sudoku: has 1 solution
– Minimal Sudoku
[Yato 03]
[Simonis 05]
• In a 9x9 Sudoku: smallest known number of givens is 17
[Royle]
– Symmetrical puzzles
• Many axes of symmetry
• Position of the givens, not their values
• Often makes the puzzle non-minimal
– Level of difficulties
• Varied ranking systems exist
• Mimimality and difficulty not related
#15 on Royle’s web site
Constraint Systems Laboratory
March 26, 2007
Reeson–Undergraduate Thesis
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Constraint Satisfaction Problem (CSP)
V1
• Given P = (V, D, C)
V2
{d}
{c, d, e, f}
V
4
V3
– V: set of variables
{a, b, d}
{a, b, c}
– D: set of their domains
– C: set of constraints (relations) restricting the
acceptable combination of values for variables
– Solution is a consistent assignment of values to
variables
• Query: find 1 solution, all solutions, etc.
Constraint Systems Laboratory
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Sudoku as a CSP
(9x9 puzzles)
• Variables are the cells
• Domains are sets {1,…,9}
• Two models
– Binary constraints: 810 all-different binary constraint between
variables
– Non-binary constraints: 27 all-different 9-ary constraints
Constraint Systems Laboratory
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Solving Sudoku as a CSP
• Search
– Builds solutions by enumerating consistent combinations
• Constraint propagation
– Removes values that do not appear in a solution
– Example, arc-consistency:
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Search
• Backtrack search
– Systematically enumerate solutions by instantiating
one variable after another
– Backtracks when a constraint is broken
– Is sound and complete (guaranteed to find solution if
one exists)
• Look-ahead
– Cleans up domain of ‘future’ variables, during search,
given current instantiations
• Forward Checking (FC): weak form of arc-consistency
• Maintaining Arc-Consistency (MAC): arc-consistency
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Forward Checking (FC)
• Forward Checking on
the binary model
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Maintaining Arc-Consistency (MAC)
• Arc Consistency on
the binary model
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Generalized Arc-Consistency (GAC)
• Operates on the nonbinary model
• Subsumes AC
• Can be done in
polynomial time for
all-different
constraints [Régin 94]
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Consistency Algorithms - SAC
• Shaving Arc-Consistency
– Binary
– Stronger than ArcConsistency
Michael Mepham Extreme #26
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Consistency Algorithms - SGAC
• Shaving Generalized
Arc-Consistency
– Non-binary
– Subsumes SAC
Michael Mepham Extreme #26
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Consistency relationships
Sets denote the values
removed by propagation
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R(1,2)C
• Takes two constraints and
• Removes any value that does not appear in a
solution to both constraints
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Consistency R(1,2)C
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Conjecture 1: R(1,2)CSGAC
 R(1,2)C is at least as strong as SGAC
Is SGAC at least as strong as R(1,2)C?
• We proved R(1,2)CSGAC when the 2 relations
overlap over one variable
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Conjecture 2: SGAC solves..
… every:
– Well-formed
– 9x9 Sudoku?
• Proof still open, but empirical evidence is strong
• Warning
– Sudoku is NP-Complete
– SGAC is polynomial
– When a 9x9 Sudoku has multiple solutions, SGAC
does not solve it
• Proof may exploit the fact that all non-binary
constraints are 1-tight
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Interactive interface
• Loading instances
• Buttons for propagation
– On individual constraints
– For consistency algorithms: AC, GAC, SAC, SGAC
– For interactive look-ahead
• Finding solutions
• Hint panel
•
•
•
•
Highlighting error
Manual domain filtering
Manual assignments
Do/Undo buttons supporting every functionality
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Buttons for constraint propagation
• Buttons allow
users to run
– AC
– GAC
• Over each
– Column
– Row
– Block
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Finding solutions
• Find the number of solutions at any point
in the game
– Creates a copy of the CSP displayed on the
board
– Runs consistency algorithms to speed up
search
– Runs exhaustive search to find all solutions
• Found Sudokus that are not well-formed
on the web and in… the LA Times
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Hint panel
• Two types of hints: Singleton & Vital
• Eight Levels (6 implemented)
– FC, AC, Single GAC, GAC, Single SAC, SAC, Single
SGAC, SGAC
• Highlight a cell with a hint
– Option to move to the next hint
– Display the number of hints
• Preliminary tests show that this new strategy
can teach people to play Sudoku effectively
and… get them quickly bored with the game
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Web-based implementation
• XML files
• Solver
• Constructor
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The way humans play
• ‘Cross-hash,’ sweep through lines, columns, and blocks
• Pencil in possible positions of values
• Generally, look for patterns, some intricate, and give
them funny names:
– Naked single, locked pairs, swordfish, medusa, etc.
• ‘Identified’ dozens of strategies
– Many fall under a unique constraint propagation technique
• But humans do not seem to be able to carry simple
inference (i.e., propagation) in a systematic way for
more than a few steps
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Impact
• Inspired a PhD student @ USC
– to use propagation to infer the applicable
constraints for automatically modeling various
types of Sudoku puzzles
• Collaborating with an MS student @ USC
on Constructor
– Inputting, storing instances
– Generalization to other Sudoku variations
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Conclusions: CP solving
• Propagation is useful for
– effectively guide humans & train them in Sudoku solving
• SAC (often), SGAC (always) can solve any well-formed
Sudoku, also noticed by Simonis
• Sudoku good illustration of the power of SAC and SGAC
– otherwise thought to be more expensive than effective
• No need for extra CP modeling as Simonis did:
– Use the simple constraint model on row, columns, blocks
– Stick to general purpose propagation algorithms
• But… Who wants to play Sudoku any more when they
know that SGAC can do it?
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Finally
• Future work:
– Use the interface to generate Sudoku puzzles
– Investigate theoretically the relationship
between
• Sudoku size
• And the consistency level necessary for solving it
• Live demonstration
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Thank you for your attention
I would be glad
to answer your questions
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