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Sudoku and Orthogonality. John Lorch Undergraduate Colloquium Fall 2008. What is a sudoku puzzle?. Sudoku ‘single number puzzle’ numbers 1-9 must appear in every row, column, and block. Typically appears in order n 2 : An n 2 × n 2 array with n × n blocks. Sudoku solution.
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Sudoku and Orthogonality John Lorch Undergraduate Colloquium Fall 2008
What is a sudoku puzzle? • Sudoku • ‘single number puzzle’ • numbers 1-9 must appear in every row, column, and block. • Typically appears in ordern2: An n2×n2 array with n×n blocks. Sudoku and Orthogonality: John Lorch
Sudoku solution • The completion of the previous puzzle: all entries are filled in. • A sudoku solution is a completed puzzle. Sudoku and Orthogonality: John Lorch
A latin squareof order n isan n×n array with n distinct symbols. Each symbol appears once in each row and column. Literature on latin squares dates to Euler (1782). Sudoku solution: a latin square with additional condition on blocks. Non-sudoku Latin square Sudoku Sudoku and Latin squares Sudoku and Orthogonality: John Lorch
Questions about sudoku • Natural questions: • How many sudoku solutions exist for a given order? • What is the minimum number of entries determining a unique completion? • What can symmetry groups tell us about sudoku? • What is known about orthogonal sudoku puzzles? • Our purpose: investigate orthogonality Sudoku and Orthogonality: John Lorch
Two latin squares are orthogonal if superimposition yields all possible ordered pairs of symbols. Orthogonality is preserved by Relabeling either square Rearrangement applied to both squares Orthogonal squares: Orthogonal Latin squares Sudoku and Orthogonality: John Lorch
Orthogonal sudoku mates • Golomb’s problem: (MAA Monthly 2006) Do there exist pairs of orthogonal sudoku solutions? • Answer: Yes • Our purpose, more specifically: • Investigate methods for producing such pairs. • Make observations on these methods. Sudoku and Orthogonality: John Lorch
Background: Transversals • Transversal: A collection of locations and corresponding entries so that each row, column, and symbol is represented exactly once. Sudoku and Orthogonality: John Lorch
Background: Transversals • Transversal Theorem: A latin square of order n has an orthogonal mate if and only if it has ndisjoint transversals. • Proof of theorem yields a method for producing an orthogonal mate. • Unfortunately, method fails to preserve sudoku block condition. Sudoku and Orthogonality: John Lorch
Method 1: Combing transversals • Consider an order 4 solution with transversal. To get orthogonal sudoku mate, we can’t apply transversal theorem directly. • Instead: Use transversals as rows (left to right combing) Sudoku and Orthogonality: John Lorch
Method 1: Combing transversals • Illustration of method with solution, yielding an orthogonal pair. Sudoku and Orthogonality: John Lorch
Method 1: Combing transversals • A different choice of transversal can yield non-orthogonal solutions: Sudoku and Orthogonality: John Lorch
Method 1: Combing transversals • Theorem: For a ‘block symmetric’ solution the combing method produces a sudoku solution. • Proof: • Rows have distinct symbols since transversals do. • Columns have distinct symbols since each (new) column is a permutation of the corresponding original column. • Blocks are rearranged and permuted, so still have distinct symbols in each block. Sudoku and Orthogonality: John Lorch
Method 1: Combing transversals Conjecture: Given a block symmetric sudoku solution, there is a choice of transversal for which the combing method produces an orthogonal sudoku mate. Sudoku and Orthogonality: John Lorch
Let α and β permute the rows and columns of block K, respectively, so that: Row i of Kα is row i+1 of K (cycle up) Column j of Kβ is column j+1 of K (cycle left) K Kα Kβ Method 2: Block Permutations Sudoku and Orthogonality: John Lorch
Method 2: Block Permutations • Theorem (Keedwell, 2007): Solutions and are orthogonal sudoku mates. One can extend the pattern to obtain orthogonal sudoku pairs of all square orders. Sudoku and Orthogonality: John Lorch
Method 2: Block permutations • Keedwell’s proof: Apply transversal theorem. Sudoku and Orthogonality: John Lorch
Another approach: Identify sudoku block locations with Zn2 Each Keedwell solution has an exponent array Exponent arrays are functions Zn2 Zn2 Z32 Method 2: Block permutations Sudoku and Orthogonality: John Lorch
Method 2: Block permutations • Theorems: • Two Keedwell sudoku solutions of order n2 are orthogonal if and only if the difference of their exponent arrays determines a bijection Zn2 Zn2 • The maximum size of an orthogonal family of sudoku solutions of order n2 is larger than or equal to p(p-1), where p is the smallest prime factor of n. Sudoku and Orthogonality: John Lorch
Applications and Connections • An easy proof of Keedwell’s Theorem: • Exponent arrays corresponding to Keedwell’s solutions are F1(i,j)=(i+j,j) and F2(i,j)=(i,i+j). Note (F2-F1)(i,j)=(-j,i) is a bijection Zn2 Zn2, so the original sudoku solutions are orthogonal.
Applications and Connections • Construction of 6 MOSu of order 9. M0 M1 M2 M3 M4 M5 Sudoku and Orthogonality: John Lorch
Applications and Connections • Observations • M1 can be achieved from M0 via combing (method 1); M2 achieved from M0 via another transversal method not discussed here. Can transversal methods be used to obtain other solutions in the collection? • Can also get 6 MOSu of order 9 by looking at the addition table for GF(9). In general, field theory and finite projective spaces can be used to determine results about orthogonality. Sudoku and Orthogonality: John Lorch
Collaborators/References • Joint with Lisa Mantini (Oklahoma State) • Principal references: • C. Colbourn and J. Dinitz, Mutually orthogonal latin squares, Journal of Statistical Planning and Inference 95 (2001), 9-48. • A. Keedwell, On sudoku squares, Bulletin of the ICA 50 (2007), 52-60. • J. Lorch, Mutually orthogonal families of linear sudoku solutions, preprint. http://www.cs.bsu.edu/homepages/jdlorch/lorchsudoku.pdf Sudoku and Orthogonality: John Lorch