2. 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/mkrebswww.calstatela.edu/faculty/mkrebs

3. Beaucoup de Sudoku (French 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/mkrebswww.calstatela.edu/faculty/mkrebs

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

5. Beaucoup de Sudoku (French for“lots”) (Spanish 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/mkrebswww.calstatela.edu/faculty/mkrebs

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

7. Beaucoup de Sudoku (French for“lots”) (Spanish for“of”) (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/mkrebswww.calstatela.edu/faculty/mkrebs

8. Beaucoup de Sudoku (French for“lots”) (Spanish for“of”) (Japanese 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/mkrebswww.calstatela.edu/faculty/mkrebs

9. 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/mkrebswww.calstatela.edu/faculty/mkrebs

10. 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/mkrebswww.calstatela.edu/faculty/mkrebs

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

12. 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/mkrebswww.calstatela.edu/faculty/mkrebs

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

15. 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/mkrebswww.calstatela.edu/faculty/mkrebs

16. 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/mkrebswww.calstatela.edu/faculty/mkrebs

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

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

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

20. The set of all column permutations which send mini- Sudokus to mini- Sudokus forms a group. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

21. What group is it? For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

22. Let’s color the columns in a different way. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

23. Let’s color the columns in a different way. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

24. Tan and lavender either switch or stay fixed. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

25. Ditto for opposite corners of a square. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

26. So the group of mini -Sudoku-preserving column is isomorphic to the group of symmetries of a square. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

27. Good exercise on isomorphisms for an undergraduate Abstract Algebra class? For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

28. In general, the group of column symmetries for an n 2 x n 2 Sudoku is an n-fold wreath product. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

46. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In fact, if the entry in the lower right is a 2, then...

47. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In fact, if the entry in the lower right is a 2, then...

48. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In fact, if the entry in the lower right is a 2, then...... take the transpose...

49. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In fact, if the entry in the lower right is a 2, then...... take the transpose...

50. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In fact, if the entry in the lower right is a 2, then...... take the transpose...then relabel.

51. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In fact, if the entry in the lower right is a 2, then...... take the transpose...then relabel.

52. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So the one with a 2 in the lower right is equivalent to the one with a 3 in the lower right.

53. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So every mini- Sudoku is equivalent to: or

54. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s fill them in. or

55. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s fill them in. or

56. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs I claim that these two are not equivalent. or

57. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs To distinguish them, we need an invariant. or

58. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Something that behaves predictably when you switch rows... or columns... or transpose... or

59. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Aha! The determinant. or

60. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s where it’s useful to think of the labels as variables. or

61. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s where it’s useful to think of the labels as variables. or

62. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or Let’s not compute the whole determinant, but rather just the “pure” 4 th degree terms.

63. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or Let’s not compute the whole determinant, but rather just the “pure” 4 th degree terms.

64. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or Up to sign and relabelling, there will still be two positive and two negative terms.

65. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or

66. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or But for the other one, it’s all positive or all negative.

67. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or These two mini- Sudokus are not equivalent.

68. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or The determinant is a complete invariant for 4 x 4 Sudokus.

69. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Question:

70. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Is the determinant is a complete invariant for 9 x 9 Sudokus? Question: