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