1. Latin Squares Jerzy Wojdyło February 17, 2006

2. Jerzy Wojdylo, Latin Squares2 Definition and Examples A Latin square is a square array in which each row and each column consists of the same set of entries without repetition. A Latin square is a square array in which each row and each column consists of the same set of entries without repetition. aabbaabcbca cababcdbcda cdab dabc

3. February 17, 2006Jerzy Wojdylo, Latin Squares3 Existence Do Latin squares exist for every n  Z + ? Do Latin squares exist for every n  Z + ? Yes. Yes.  Consider the addition table (the Cayley table) of the group Z n.  Or, more generally, consider the multiplication table of an n-element quasigroup.

4. February 17, 2006Jerzy Wojdylo, Latin Squares4 Latin Squares and Quasigroups A quasigroup is is a nonempty set Q with operation · : Q  Q (multiplication) such that in the equation r · c = s the values of any two variables determine the third one uniquely. A quasigroup is is a nonempty set Q with operation · : Q  Q (multiplication) such that in the equation r · c = s the values of any two variables determine the third one uniquely. It is like a group, but associativity and the unit element are optional. It is like a group, but associativity and the unit element are optional.

5. February 17, 2006Jerzy Wojdylo, Latin Squares5 Latin Squares and Quasigroups The uniqueness guarantees no repetitions of symbols s in each row r and each column c. The uniqueness guarantees no repetitions of symbols s in each row r and each column c. ·0123 00231 13102 22310 31023

6. February 17, 2006Jerzy Wojdylo, Latin Squares6 Operations on Latin Squares Isotopism of a Latin square L is a Isotopism of a Latin square L is a  permutation of its rows,  permutation of its columns,  permutation of its symbols. (These permutations do not have to be the same.) (These permutations do not have to be the same.) L is reduced iff its first row is [1, 2, …, n] and its first column is [1, 2, …, n] T. L is reduced iff its first row is [1, 2, …, n] and its first column is [1, 2, …, n] T. L is normal iff its first row is [1, 2, …, n]. L is normal iff its first row is [1, 2, …, n].

7. February 17, 2006Jerzy Wojdylo, Latin Squares7 Enumeration How many Latin squares (Latin rectangles) are there? How many Latin squares (Latin rectangles) are there? If order  11 Brendan D. McKay, Ian M. Wanless, “The number of Latin squares of order eleven” 2004(?) (show the table on page 5) http://en.wikipedia.org/wiki/Latin_square#The_number_of_Latin_squares If order  11 Brendan D. McKay, Ian M. Wanless, “The number of Latin squares of order eleven” 2004(?) (show the table on page 5) http://en.wikipedia.org/wiki/Latin_square#The_number_of_Latin_squares http://en.wikipedia.org/wiki/Latin_square#The_number_of_Latin_squares Order 12, 13, … open problem. Order 12, 13, … open problem.

8. February 17, 2006Jerzy Wojdylo, Latin Squares8 Orthogonal Latin Squares Two n  n Latin squares L=[l ij ] and M =[m ij ] are orthogonal iff the n 2 pairs (l ij, m ij ) are all different. Two n  n Latin squares L=[l ij ] and M =[m ij ] are orthogonal iff the n 2 pairs (l ij, m ij ) are all different. abc bca cababccab bca

9. February 17, 2006Jerzy Wojdylo, Latin Squares9 Orthogonal LS - Useful Property Theorem Two Latin squares are orthogonal iff their normal forms are orthogonal. (You can symbols so both LS have the first row [1, 2, …, n]) Theorem Two Latin squares are orthogonal iff their normal forms are orthogonal. (You can symbols so both LS have the first row [1, 2, …, n]) No two 2  2 Latin squares are orthogonal. No two 2  2 Latin squares are orthogonal. 12 21

10. February 17, 2006Jerzy Wojdylo, Latin Squares10 Orthogonal Latin Squares This 4  4 Latin square does not have an orthogonal mate. This 4  4 Latin square does not have an orthogonal mate. 1234 2341 3412 41231234

11. February 17, 2006Jerzy Wojdylo, Latin Squares11 Orthogonal LS - History 1782 Leonhard Euler 1782 Leonhard Euler  The problem of 36 officers, 6 ranks, 6 regiments. His conclusion: No two 6  6 LS are orthogonal.  Additional conjecture: no two n  n LS are orthogonal, where n  Z +, n  2 (mod 4). 1900 G. Tarry verified the case n = 6. 1900 G. Tarry verified the case n = 6.

12. February 17, 2006Jerzy Wojdylo, Latin Squares12 Orthogonal LS – History (cd) 1960 R.C. Bose, S.S. Shrikhande, E.T. Parker, Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture, Canadian Journal of Mathematics, vol. 12 (1960), pp. 189-203. 1960 R.C. Bose, S.S. Shrikhande, E.T. Parker, Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture, Canadian Journal of Mathematics, vol. 12 (1960), pp. 189-203. There exists a pair of orthogonal LS for all n  Z +, with exception of n = 2 and n = 6. There exists a pair of orthogonal LS for all n  Z +, with exception of n = 2 and n = 6.

13. February 17, 2006Jerzy Wojdylo, Latin Squares13 Mutually Orthogonal LS (MOLS) A set of LS that are pairwise orthogonal is called a set of mutually orthogonal Latin squares (MOLS). A set of LS that are pairwise orthogonal is called a set of mutually orthogonal Latin squares (MOLS). Theorem The largest number of n  n MOLS is n  1. Theorem The largest number of n  n MOLS is n  1.

14. February 17, 2006Jerzy Wojdylo, Latin Squares14 Mutually Orthogonal LS (MOLS) Proof (by contradiction) Proof (by contradiction) Suppose we have n MOLS: … … … … … … L 1 L i L j L n 12…n12…n12…n12…n

15. February 17, 2006Jerzy Wojdylo, Latin Squares15 MOLS Theorem If n = p, prime, then there are n  1 n  n-MOLS. Theorem If n = p, prime, then there are n  1 n  n-MOLS. Proof Construction of L k =[a k ij ], k =1, 2, …, n  1: a k ij = ki + j (mod n). Proof Construction of L k =[a k ij ], k =1, 2, …, n  1: a k ij = ki + j (mod n). Corollary If n=p t, p prime, then there are n  1 n  n-MOLS. Corollary If n=p t, p prime, then there are n  1 n  n-MOLS. Open problem If there are n  1 n  n-MOLS, then n = p t, p prime. Open problem If there are n  1 n  n-MOLS, then n = p t, p prime.

16. February 17, 2006Jerzy Wojdylo, Latin Squares16 Latin Rectangle A p  q Latin rectangle with entries in {1, 2, …, n} is a p  q matrix with entries in {1, 2, …, n} with no repeated entry in a row or column. A p  q Latin rectangle with entries in {1, 2, …, n} is a p  q matrix with entries in {1, 2, …, n} with no repeated entry in a row or column. (3,4,5) Latin rectangle (3,4,5) Latin rectangle 1345 3512 5134

17. February 17, 2006Jerzy Wojdylo, Latin Squares17 Completion Problems When can a p  q Latin rectangle with entries in {1, 2, …, n} be completed to a n  n Latin square? When can a p  q Latin rectangle with entries in {1, 2, …, n} be completed to a n  n Latin square? 1345 3512 5134 12344312

18. February 17, 2006Jerzy Wojdylo, Latin Squares18 Completion Problems The good: The good: 1234 4312 2143 3421

19. February 17, 2006Jerzy Wojdylo, Latin Squares19 Completion Problems The bad: The bad: Where to put “2” in the last column? 1345 3512 5134

20. February 17, 2006Jerzy Wojdylo, Latin Squares20 Completion Theorems Theorem Let p < n. Any p  n Latin rectangle with entries in {1, 2, …, n} can be completed to a n  n Latin square. Theorem Let p < n. Any p  n Latin rectangle with entries in {1, 2, …, n} can be completed to a n  n Latin square. The proof uses Hall’s marriage theorem or transversals to complete the bottom n  p rows. The construction fills one row at a time. The proof uses Hall’s marriage theorem or transversals to complete the bottom n  p rows. The construction fills one row at a time.

21. February 17, 2006Jerzy Wojdylo, Latin Squares21 Completion Theorems Theorem Let p, q < n. A p  q Latin rectangle R with entries in {1, 2, …, n} can be completed to a n  n Latin square iff R(t), the number of occurrences of t in R, satisfies R(t)  p + q  n for each t with 1  t  n. Theorem Let p, q < n. A p  q Latin rectangle R with entries in {1, 2, …, n} can be completed to a n  n Latin square iff R(t), the number of occurrences of t in R, satisfies R(t)  p + q  n for each t with 1  t  n.

22. February 17, 2006Jerzy Wojdylo, Latin Squares22 Completion Theorems From last slide: R(t)  p + q  n. From last slide: R(t)  p + q  n. Let t = 5. Then R(5) = 1 and p+q  n = 4+4  6 = 2. But 1  2, so R cannot be completed to a Latin square. 6123 5631 1362 3246

23. February 17, 2006Jerzy Wojdylo, Latin Squares23 Completion Problems The ugly (?) a. k. a. sudoku The ugly (?) a. k. a. sudoku 971 74265 1894 2859 1236 4512 7412 32985 657

24. February 17, 2006Jerzy Wojdylo, Latin Squares24 Completion Problems The ugly (?) a. k. a. sudoku The ugly (?) a. k. a. sudoku 963467218 874132965 612895473 236789591 198253746 745614382 587341629 321976854 469528137

25. February 17, 2006Jerzy Wojdylo, Latin Squares25 Sudoku History: History:  http://en.wikipedia.org/wiki/Sudoku http://en.wikipedia.org/wiki/Sudoku  Robin Wilson, The Sudoku Epidemic, MAA Focus, January 2006.  http://sudoku.com/ http://sudoku.com/  Google (2/15/2006) about 20,300,000 results for sudoku.

26. February 17, 2006Jerzy Wojdylo, Latin Squares26 Mathematics of Sudoku Bertram Felgenhauer and Frazer Jarvis: Bertram Felgenhauer and Frazer Jarvis:  There are 6,670,903,752,021,072,936,960 Sudoku grids. Ed Russell and Frazer Jarvis: Ed Russell and Frazer Jarvis:  There are 5,472,730,538 essentially different Sudoku grids. http://www.afjarvis.staff.shef.ac.uk/sudoku/ http://www.afjarvis.staff.shef.ac.uk/sudoku/ http://www.afjarvis.staff.shef.ac.uk/sudoku/

27. February 17, 2006Jerzy Wojdylo, Latin Squares27 Uniqueness of Sudoku Completion Maximal number of givens while solution is not unique: 81  4 = 77. Maximal number of givens while solution is not unique: 81  4 = 77. ?? ??

28. February 17, 2006Jerzy Wojdylo, Latin Squares28 Uniqueness of Sudoku Completion Minimal number of givens which force a unique solution – open problem. Minimal number of givens which force a unique solution – open problem. So far: So far:  the lowest number yet found for the standard variation without a symmetry constraint is 17,  and 18 with the givens in rotationally symmetric cells.

29. February 17, 2006Jerzy Wojdylo, Latin Squares29 Example of Small Sudoku 1 4 2 547 83 19 342 51 86

30. February 17, 2006Jerzy Wojdylo, Latin Squares30 Example of Small Sudoku 693784512 487512936 125963874 932651487 568247391 741398625 319475268 856129743 274836159

31. February 17, 2006Jerzy Wojdylo, Latin Squares31 More Small Sudoku Grids Sudoku grids with 17 givens http://www.csse.uwa.edu.au/~gordon/sudok umin.php Sudoku grids with 17 givens http://www.csse.uwa.edu.au/~gordon/sudok umin.php http://www.csse.uwa.edu.au/~gordon/sudok umin.php http://www.csse.uwa.edu.au/~gordon/sudok umin.php Need help solving sudoku? Try: http://www.sudokusolver.co.uk/ Need help solving sudoku? Try: http://www.sudokusolver.co.uk/ http://www.sudokusolver.co.uk/

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