1. Sudoku and Orthogonality
John Lorch
Undergraduate Colloquium
Fall 2008

2. Sudoku and Orthogonality: John Lorch
What is a sudoku puzzle?
Sudoku
‘single number puzzle’
numbers 1-9 must appear in every row, column, and block.
Typically appears in order n2: An n2×n2 array with n×n blocks. 1 7 9 6 8 2 5 3 4
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Sudoku and Orthogonality: John Lorch

3. 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. 1 5 7 4 3 8 9 2 6
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Sudoku and Orthogonality: John Lorch

4. Sudoku and Latin squares
A latin square of order n is an 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 1 2 3 4 1 2 4 3
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Sudoku and Orthogonality: John Lorch

5. 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
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Sudoku and Orthogonality: John Lorch

6. Orthogonal Latin squares
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: 4 3 2 1 2 1 4 3 42 31 24 13 34 43 12 21 23 14 41 32 11 22 33 44
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Sudoku and Orthogonality: John Lorch

7. 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.
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Sudoku and Orthogonality: John Lorch

8. Background: Transversals
Transversal: A collection of locations and corresponding entries so that each row, column, and symbol is represented exactly once. 1 2 5 3 4 8 6 7
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Sudoku and Orthogonality: John Lorch

9. Background: Transversals
Transversal Theorem: A latin square of order n has an orthogonal mate if and only if it has n disjoint transversals.
Proof of theorem yields a method for producing an orthogonal mate.
Unfortunately, method fails to preserve sudoku block condition. 2 1 4 3 1 2 3 4 21 31 11 41
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Sudoku and Orthogonality: John Lorch

10. Method 1: Combing transversals2 4 3 1 4 3 2
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) 11 24 43 32 33 42 21 14 44 31 12 23 22 13 34 41
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Sudoku and Orthogonality: John Lorch

11. Method 1: Combing transversals
Illustration of method with solution, yielding an orthogonal pair. 1 2 5 3 4 8 6 7 4 8 3 7 2 6 1 5
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Sudoku and Orthogonality: John Lorch

12. Method 1: Combing transversals
A different choice of transversal can yield non-orthogonal solutions: 1 2 5 3 4 8 6 7 7 5 6 4 2 3 1 8
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Sudoku and Orthogonality: John Lorch

13. 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.
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Sudoku and Orthogonality: John Lorch

14. 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.
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Sudoku and Orthogonality: John Lorch

15. Method 2: Block Permutations
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 1 2 3 4 5 6 7 8 3 4 5 6 7 8 1 2 1 2 4 5 3 7 8 6
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Sudoku and Orthogonality: John Lorch

16. 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. K Kα
Kα2
Kαβ
Kα2β Kβ
Kα2β2
Kβ2
Kαβ2 K
Kαβ
Kα2β2 Kβ
Kαβ2
Kα2
Kβ2 Kα
Kα2β
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Sudoku and Orthogonality: John Lorch

17. Method 2: Block permutations
Keedwell’s proof: Apply transversal theorem. 1 2 3 4 5 6 7 8 1 2 4 5 3 8 6 7
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Sudoku and Orthogonality: John Lorch

18. Method 2: Block permutations
Another approach:
Identify sudoku block locations with Zn2
Each Keedwell solution has an exponent array
Exponent arrays are functions
Zn2  Zn2
Z32
(0,0)
(0,1)
(0,2)
(1,0)
(1,1)
(1,2)
(2,0)
(2,1)
(2,2) K Kα
Kα2
Kαβ
Kα2β Kβ
Kα2β2
Kβ2
Kαβ2
(0,0)
(1,0)
(2,0)
(1,1)
(2,1)
(0,1)
(2,2)
(0,2)
(1,2)
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Sudoku and Orthogonality: John Lorch

19. 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.
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Sudoku and Orthogonality: John Lorch

20. 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.

21. Applications and Connections
Construction of 6 MOSu of order 9.
M M M2
M M M5 K
Kαβ2
Kα2β K Kα
Kα2 Kβ
Kαβ
Kα2β
Kβ2
Kαβ2
Kα2β2 K
Kαβ
Kα2β2
Kαβ2
Kα2 Kβ
Kα2β
Kβ2 Kα K
Kα2β2
Kαβ
Kβ2
Kα2β Kα Kβ
Kα2
Kαβ2 K
Kα2 Kα
Kα2β2
Kαβ2
Kβ2
Kαβ Kβ
Kα2β K
Kα2β
Kαβ2
Kαβ
Kβ2
Kα2
Kα2β2 Kα Kβ
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Sudoku and Orthogonality: John Lorch

22. 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.
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Sudoku and Orthogonality: John Lorch

23. 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),
J. Lorch, Mutually orthogonal families of linear sudoku solutions, preprint.
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Sudoku and Orthogonality: John Lorch