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

http://www.calstatela.edu/faculty/mkrebs

Beaucoup

de

Sudoku

(French for




Beaucoup

de

Sudoku

(French for “lots”)




Beaucoup

de

Sudoku


(Spanish for




Beaucoup

de

Sudoku


(Spanish for “of”)




Beaucoup

de

Sudoku



(Japanese for




Beaucoup

de

Sudoku



(Japanese for “Sudoku”)




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.



The digits 1 through 9 are just labels.




They could just as well be variables . . .




They could just as well be variables . . .

. . . or . . .



To keep things simple, we’ll consider the smallercase of 4 by 4 Sudokus; we call these mini-Sudokus.



There are several obvious ways to obtain a newmini-Sudoku from an old one.



For example, you can switch the first two columns.



The set of all column permutations which sendmini-Sudokus to mini-Sudokus forms a group.



What group is it?



Let’s color the columns in a different way.



Tan and lavender either switch or stay fixed.



Ditto for opposite corners of a square.



So the group of mini-Sudoku-preserving column isisomorphic to the group of symmetries of a square.



Good exercise on isomorphisms for anundergraduate Abstract Algebra class?



In general, the group of column symmetries foran n2 x n2 Sudoku is an n-fold wreath product.




Of course, in addition to permuting columns, wecan also permute rows . . .



. . . “transpose” the mini-Sudoku . . .



. . . or relabel entries.



We say two mini-Sudokus are equivalent if youcan get from one to the other via a finite sequenceof row/column permutations, transpositions, andrelabellings.



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?



Given any mini-Sudoku, we can always apply arelabelling to get a new mini-Sudoku of this form:



Then apply row and column permutations to get:



The mini-Sudoku is then determined by this entry:



So every mini-Sudoku is equivalent to one of threemini-Sudokus.



In fact, if the entry in the lower right is a 2, then . . .




. . . take the transpose . . .




. . . take the transpose . . . then relabel.



So the one with a 2 in the lower right is equivalentto the one with a 3 in the lower right.



So every mini-Sudoku is equivalent to:

or



Let’s fill them in.

or



I claim that these two are not equivalent.

or



To distinguish them, we need an invariant.

or



Something that behaves predictably when youswitch rows . . . or columns . . . or transpose . . .

or



Aha! The determinant.

or



Here’s where it’s useful to think of the labels asvariables.

or



or

Let’s not compute the whole determinant, butrather just the “pure” 4th degree terms.


or

Up to sign and relabelling, there will stillbe two positive and two negative terms.


or


or

But for the other one, it’s all positive orall negative.


or

These two mini-Sudokus are not equivalent.


or

The determinant is a complete invariant for4 x 4 Sudokus.


Question:


Is the determinant is a completeinvariant for 9 x 9 Sudokus?

Question:

Beaucoup de Sudoku Mike Krebs, Cal State LA(joint work with C. Arcos and G. Brookfield) For...

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