Beaucoup de Sudoku Mike Krebs, Cal State LA(joint work with C. Arcos and G. Brookfield) For slideshow: click “Research and Talks” from
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
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
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
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
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
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
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
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
The digits 1 through 9 are just labels. For slideshow: click “Research and Talks” from
The digits 1 through 9 are just labels. They could just as well be variables... For slideshow: click “Research and Talks” from
The digits 1 through 9 are just labels. They could just as well be variables or... For slideshow: click “Research and Talks” from
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
There are several obvious ways to obtain a new mini- Sudoku from an old one. For slideshow: click “Research and Talks” from
For example, you can switch the first two columns. For slideshow: click “Research and Talks” from
For example, you can switch the first two columns. For slideshow: click “Research and Talks” from
For example, you can switch the first two columns. For slideshow: click “Research and Talks” from
The set of all column permutations which send mini- Sudokus to mini- Sudokus forms a group. For slideshow: click “Research and Talks” from
What group is it? For slideshow: click “Research and Talks” from
Let’s color the columns in a different way. For slideshow: click “Research and Talks” from
Let’s color the columns in a different way. For slideshow: click “Research and Talks” from
Tan and lavender either switch or stay fixed. For slideshow: click “Research and Talks” from
Ditto for opposite corners of a square. For slideshow: click “Research and Talks” from
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
Good exercise on isomorphisms for an undergraduate Abstract Algebra class? For slideshow: click “Research and Talks” from
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
For slideshow: click “Research and Talks” from Of course, in addition to permuting columns, we can also permute rows...
For slideshow: click “Research and Talks” from Of course, in addition to permuting columns, we can also permute rows...
For slideshow: click “Research and Talks” from Of course, in addition to permuting columns, we can also permute rows...
For slideshow: click “Research and Talks” from “transpose” the mini- Sudoku...
For slideshow: click “Research and Talks” from “transpose” the mini- Sudoku...
For slideshow: click “Research and Talks” from or relabel entries.
For slideshow: click “Research and Talks” from or relabel entries.
For slideshow: click “Research and Talks” from or relabel entries.
For slideshow: click “Research and Talks” from 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.
For slideshow: click “Research and Talks” from 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.
For slideshow: click “Research and Talks” from 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?
For slideshow: click “Research and Talks” from 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 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 Then apply row and column permutations to get:
For slideshow: click “Research and Talks” from The mini- Sudoku is then determined by this entry:
For slideshow: click “Research and Talks” from The mini- Sudoku is then determined by this entry:
For slideshow: click “Research and Talks” from So every mini- Sudoku is equivalent to one of three mini- Sudokus.
For slideshow: click “Research and Talks” from In fact, if the entry in the lower right is a 2, then...
For slideshow: click “Research and Talks” from In fact, if the entry in the lower right is a 2, then...
For slideshow: click “Research and Talks” from In fact, if the entry in the lower right is a 2, then take the transpose...
For slideshow: click “Research and Talks” from In fact, if the entry in the lower right is a 2, then take the transpose...
For slideshow: click “Research and Talks” from In fact, if the entry in the lower right is a 2, then take the transpose...then relabel.
For slideshow: click “Research and Talks” from In fact, if the entry in the lower right is a 2, then take the transpose...then relabel.
For slideshow: click “Research and Talks” from So the one with a 2 in the lower right is equivalent to the one with a 3 in the lower right.
For slideshow: click “Research and Talks” from So every mini- Sudoku is equivalent to: or
For slideshow: click “Research and Talks” from Let’s fill them in. or
For slideshow: click “Research and Talks” from Let’s fill them in. or
For slideshow: click “Research and Talks” from I claim that these two are not equivalent. or
For slideshow: click “Research and Talks” from To distinguish them, we need an invariant. or
For slideshow: click “Research and Talks” from Something that behaves predictably when you switch rows... or columns... or transpose... or
For slideshow: click “Research and Talks” from Aha! The determinant. or
For slideshow: click “Research and Talks” from Here’s where it’s useful to think of the labels as variables. or
For slideshow: click “Research and Talks” from Here’s where it’s useful to think of the labels as variables. or
For slideshow: click “Research and Talks” from or Let’s not compute the whole determinant, but rather just the “pure” 4 th degree terms.
For slideshow: click “Research and Talks” from or Let’s not compute the whole determinant, but rather just the “pure” 4 th degree terms.
For slideshow: click “Research and Talks” from or Up to sign and relabelling, there will still be two positive and two negative terms.
For slideshow: click “Research and Talks” from or
For slideshow: click “Research and Talks” from or But for the other one, it’s all positive or all negative.
For slideshow: click “Research and Talks” from or These two mini- Sudokus are not equivalent.
For slideshow: click “Research and Talks” from or The determinant is a complete invariant for 4 x 4 Sudokus.
For slideshow: click “Research and Talks” from Question:
For slideshow: click “Research and Talks” from Is the determinant is a complete invariant for 9 x 9 Sudokus? Question: