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Beaucoup

Beaucoup. de. Sudoku. Mike Krebs, Cal State LA. (joint work with C. Arcos and G. Brookfield). For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs. Beaucoup. (French for. de. Sudoku. Mike Krebs, Cal State LA. (joint work with C. Arcos and G. Brookfield).

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Beaucoup

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  1. Beaucoup de Sudoku Mike Krebs, Cal State LA (joint work with C. Arcos and G. Brookfield) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  2. Beaucoup (French for de Sudoku Mike Krebs, Cal State LA (joint work with C. Arcos and G. Brookfield) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  3. Beaucoup (French for “lots”) de Sudoku Mike Krebs, Cal State LA (joint work with C. Arcos and G. Brookfield) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  4. Beaucoup (French for “lots”) de (Spanish for Sudoku Mike Krebs, Cal State LA (joint work with C. Arcos and G. Brookfield) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  5. Beaucoup (French for “lots”) de (Spanish for “of”) Sudoku Mike Krebs, Cal State LA (joint work with C. Arcos and G. Brookfield) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  6. Beaucoup (French for “lots”) de (Spanish for “of”) Sudoku (Japanese for Mike Krebs, Cal State LA (joint work with C. Arcos and G. Brookfield) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  7. Beaucoup (French for “lots”) de (Spanish for “of”) Sudoku (Japanese for “数字は独身に限る”) Mike Krebs, Cal State LA (joint work with C. Arcos and G. Brookfield) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  8. 数字は独身に限る For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  9. 数字は独身に限る suji wa dokushin ni kagiru For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  10. 数字は独身に限る suji wa dokushin ni kagiru the digits must be single For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  11. 数字は独身に限る 数独 For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  12. 数字は独身に限る suji wa dokushin ni kagiru 数独 su doku For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  13. 数字は独身に限る suji wa dokushin ni kagiru the digits must be single 数独 su doku single digits For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  14. A Sudoku is a 9 by 9 grid of digits in which every row, every column, and every 3 by 3 box with thick borders contains each digit from 1 to 9 exactly once. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  15. A Sudoku is a 9 by 9 grid of digits in which every row, every column, and every 3 by 3 box with thick borders contains each digit from 1 to 9 exactly once. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  16. Typically, one is given some of the entries and must fill in the remaining ones. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  17. Typically, one is given some of the entries and must fill in the remaining ones. There are many interesting strategies for how to do this. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  18. Typically, one is given some of the entries and must fill in the remaining ones. There are many interesting strategies for how to do this. Today, we will not be learning about any of them. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  19. (Side comment: the Sudoku on the right has only 17 “filled-in” entries, yet it has a unique solution. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  20. (Side comment: the Sudoku on the right has only 17 “filled-in” entries, yet it has a unique solution. It is conjectured that this is the smallest possible number of “givens” one can have and still have a unique solution.) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  21. The digits 1 through 9 are just labels. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  22. The digits 1 through 9 are just labels. They could just as well be variables . . . For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  23. The digits 1 through 9 are just labels. They could just as well be variables . . . . . . or colors . . . For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  24. The digits 1 through 9 are just labels. They could just as well be variables . . . . . . or colors . . . . . . or . . . For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  25. Divizio Unu: Ekvivalenta malgranda Sudokoj For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  26. Divizio Unu: Ekvivalenta malgranda Sudokoj Part One: Equivalent mini-Sudokus For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  27. To keep things simple, we’ll consider the smaller case of 4 by 4 Sudokus; we call these mini-Sudokus. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  28. There are several obvious ways to obtain a new mini-Sudoku from an old one. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  29. For example, you can switch the first two columns. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  30. For example, you can switch the first two columns. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  31. You may not switch columns willy-nilly; tan and lavender either switch or stay fixed. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  32. Of course, in addition to permuting columns, we can also permute rows . . . For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  33. Of course, in addition to permuting columns, we can also permute rows . . . For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  34. . . . “transpose” the mini-Sudoku . . . For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  35. . . . “transpose” the mini-Sudoku . . . For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  36. . . . or relabel entries. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  37. . . . or relabel entries. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  38. We say two mini-Sudokus are equivalent if you can get from one to the other via a finite sequenceof row/column permutations, transpositions, and relabellings. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  39. We say two mini-Sudokus are equivalent if you can get from one to the other via a finite sequenceof row/column permutations, transpositions, and relabellings. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  40. We say two mini-Sudokus are equivalent if you can get from one to the other via a finite sequenceof row/column permutations, transpositions, and relabellings. Are all mini-Sudokus equivalent? For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  41. Given any mini-Sudoku, we can always apply a relabelling to get a new mini-Sudoku of this form: For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  42. Given any mini-Sudoku, we can always apply a relabelling to get a new mini-Sudoku of this form: For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  43. Then apply row and column permutations to get: For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  44. The mini-Sudoku is then determined by this entry: For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  45. So, every mini-Sudoku is equivalent to one of three mini-Sudokus. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  46. And now . . . a brief digression on counting Sudokus. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  47. 4! For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

  48. 4!·2·2 For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

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