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Latin Squares

Latin Squares. Jerzy Wojdy ł o February 17, 2006. 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. Existence. Do Latin squares exist for every n  Z + ? Yes.

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Latin Squares

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  1. Latin Squares Jerzy Wojdyło February 17, 2006

  2. 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. Jerzy Wojdylo, Latin Squares

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

  4. 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. • It is like a group, but associativity and the unit element are optional. Jerzy Wojdylo, Latin Squares

  5. Latin Squares and Quasigroups • The uniqueness guarantees no repetitions of symbols s in each row r and each column c. Jerzy Wojdylo, Latin Squares

  6. Operations on Latin Squares • 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.) • 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]. Jerzy Wojdylo, Latin Squares

  7. Enumeration • How many Latin squares (Latin rectangles) are there? • If order  11Brendan 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 • Order 12, 13, … open problem. Jerzy Wojdylo, Latin Squares

  8. Orthogonal Latin Squares • Two nn Latin squares L=[lij] and M =[mij] are orthogonal iff the n2 pairs (lij, mij) are all different. Jerzy Wojdylo, Latin Squares

  9. Orthogonal LS - Useful Property • TheoremTwo 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. Jerzy Wojdylo, Latin Squares

  10. Orthogonal Latin Squares • This 44 Latin square does not have an orthogonal mate. Jerzy Wojdylo, Latin Squares

  11. Orthogonal LS - History • 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. Jerzy Wojdylo, Latin Squares

  12. 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. • There exists a pair of orthogonal LS for all nZ+, with exception of n = 2 and n = 6. Jerzy Wojdylo, Latin Squares

  13. Mutually Orthogonal LS (MOLS) • A set of LS that are pairwise orthogonal is called a set of mutually orthogonal Latin squares (MOLS). • TheoremThe largest number of nn MOLS is n1. Jerzy Wojdylo, Latin Squares

  14. Mutually Orthogonal LS (MOLS) • Proof (by contradiction) Suppose we have n MOLS: … … … L1 Li Lj Ln Jerzy Wojdylo, Latin Squares

  15. MOLS • TheoremIf n = p, prime, then there are n1nn-MOLS. • ProofConstruction of Lk=[akij], k =1, 2, …, n1: akij = ki + j (mod n).  • CorollaryIf n=pt, p prime, then there are n1nn-MOLS. • Open problemIf there are n1nn-MOLS, then n = pt, p prime. Jerzy Wojdylo, Latin Squares

  16. Latin Rectangle • A pqLatin 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 Jerzy Wojdylo, Latin Squares

  17. Completion Problems • When can a pqLatin rectangle with entries in {1, 2, …, n} be completed to a nnLatin square? Jerzy Wojdylo, Latin Squares

  18. Completion Problems • The good: Jerzy Wojdylo, Latin Squares

  19. Completion Problems • The bad: Where to put “2” in the last column? Jerzy Wojdylo, Latin Squares

  20. Completion Theorems • TheoremLet p < n. Any pnLatin rectangle with entries in {1, 2, …, n} can be completed to a nnLatin 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. Jerzy Wojdylo, Latin Squares

  21. Completion Theorems • TheoremLet p, q < n. A pqLatin rectangle R with entries in {1, 2, …, n} can be completed to a nnLatin square iff R(t), the number of occurrences of t in R, satisfiesR(t)  p + q  nfor each t with1  t  n. Jerzy Wojdylo, Latin Squares

  22. Completion Theorems • 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. Jerzy Wojdylo, Latin Squares

  23. Completion Problems • The ugly (?)a. k. a. sudoku Jerzy Wojdylo, Latin Squares

  24. Completion Problems • The ugly (?)a. k. a. sudoku Jerzy Wojdylo, Latin Squares

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

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

  27. Uniqueness of Sudoku Completion • Maximal number of givens while solution is not unique: 81  4 = 77. Jerzy Wojdylo, Latin Squares

  28. Uniqueness of Sudoku Completion • Minimal number of givens which force a unique solution – open problem. • 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. Jerzy Wojdylo, Latin Squares

  29. Example of Small Sudoku Jerzy Wojdylo, Latin Squares

  30. Example of Small Sudoku Jerzy Wojdylo, Latin Squares

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

  32. The End

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