Post on 20-Dec-2015
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
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/mkrebs
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/mkrebs
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/mkrebs
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/mkrebs
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/mkrebs
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/mkrebs
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.
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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.
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The digits 1 through 9 are just labels.
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The digits 1 through 9 are just labels.
They could just as well be variables . . .
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The digits 1 through 9 are just labels.
They could just as well be variables . . .
. . . or . . .
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To keep things simple, we’ll consider the smallercase of 4 by 4 Sudokus; we call these mini-Sudokus.
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There are several obvious ways to obtain a newmini-Sudoku from an old one.
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For example, you can switch the first two columns.
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For example, you can switch the first two columns.
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For example, you can switch the first two columns.
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The set of all column permutations which sendmini-Sudokus to mini-Sudokus forms a group.
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What group is it?
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Let’s color the columns in a different way.
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Let’s color the columns in a different way.
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Tan and lavender either switch or stay fixed.
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Ditto for opposite corners of a square.
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So the group of mini-Sudoku-preserving column isisomorphic to the group of symmetries of a square.
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Good exercise on isomorphisms for anundergraduate Abstract Algebra class?
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In general, the group of column symmetries foran n2 x n2 Sudoku is an n-fold wreath product.
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Of course, in addition to permuting columns, wecan also permute rows . . .
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Of course, in addition to permuting columns, wecan also permute rows . . .
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Of course, in addition to permuting columns, wecan also permute rows . . .
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. . . “transpose” the mini-Sudoku . . .
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. . . “transpose” the mini-Sudoku . . .
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. . . or relabel entries.
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. . . or relabel entries.
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. . . or relabel entries.
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We say two mini-Sudokus are equivalent if youcan get from one to the other via a finite sequenceof row/column permutations, transpositions, andrelabellings.
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We say two mini-Sudokus are equivalent if youcan get from one to the other via a finite sequenceof row/column permutations, transpositions, andrelabellings.
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs
We say two mini-Sudokus are equivalent if youcan get from one to the other via a finite sequenceof row/column permutations, transpositions, andrelabellings.
Are all mini-Sudokus equivalent?
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Given any mini-Sudoku, we can always apply arelabelling to get a new mini-Sudoku of this form:
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Given any mini-Sudoku, we can always apply arelabelling to get a new mini-Sudoku of this form:
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Then apply row and column permutations to get:
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The mini-Sudoku is then determined by this entry:
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The mini-Sudoku is then determined by this entry:
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So every mini-Sudoku is equivalent to one of threemini-Sudokus.
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In fact, if the entry in the lower right is a 2, then . . .
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In fact, if the entry in the lower right is a 2, then . . .
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In fact, if the entry in the lower right is a 2, then . . .
. . . take the transpose . . .
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In fact, if the entry in the lower right is a 2, then . . .
. . . take the transpose . . .
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In fact, if the entry in the lower right is a 2, then . . .
. . . take the transpose . . . then relabel.
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In fact, if the entry in the lower right is a 2, then . . .
. . . take the transpose . . . then relabel.
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So the one with a 2 in the lower right is equivalentto the one with a 3 in the lower right.
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So every mini-Sudoku is equivalent to:
or
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Let’s fill them in.
or
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Let’s fill them in.
or
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I claim that these two are not equivalent.
or
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To distinguish them, we need an invariant.
or
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Something that behaves predictably when youswitch rows . . . or columns . . . or transpose . . .
or
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Aha! The determinant.
or
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Here’s where it’s useful to think of the labels asvariables.
or
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Here’s where it’s useful to think of the labels asvariables.
or
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or
Let’s not compute the whole determinant, butrather just the “pure” 4th degree terms.
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or
Let’s not compute the whole determinant, butrather just the “pure” 4th degree terms.
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or
Up to sign and relabelling, there will stillbe two positive and two negative terms.
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or
But for the other one, it’s all positive orall negative.
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or
These two mini-Sudokus are not equivalent.
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or
The determinant is a complete invariant for4 x 4 Sudokus.