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.

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

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

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

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

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

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

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 (Japanese for “ 数字は独身に限る ”)

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 数字は独身に限る

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 数字は独身に限る suji wa dokushin ni kagiru

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 数字は独身に限る the digits must be single suji wa dokushin ni kagiru

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 数字は独身に限る数独

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 数字は独身に限る数独 su doku suji wa dokushin ni kagiru

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 数字は独身に限る the digits must be single 数独 suji wa dokushin ni kagiru su doku single digits

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

Typically, one is given some of the entries and must fill in the remaining ones. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

Typically, one is given some of the entries and must fill in the remaining ones. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs There are many interesting strategies for how to do this.

Typically, one is given some of the entries and must fill in the remaining ones. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs There are many interesting strategies for how to do this. Today, we will not be learning about any of them.

(Side comment: the Sudoku on the right has only 17 “filled-in” entries, yet it has a unique solution. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

(Side comment: the Sudoku on the right has only 17 “filled-in” entries, yet it has a unique solution. It is conjectured that this is the smallest possible number of “givens” one can have and still have a unique solution.) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

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

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

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

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

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Divizio Unu: Ekvivalenta malgranda Sudokoj

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Divizio Unu: Part One: Equivalent mini- Sudokus Ekvivalenta malgranda Sudokoj

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

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

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

You may not switch columns willy-nilly; tan and lavender either switch or stay fixed. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

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...

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

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

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.

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?

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:

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

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:

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.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs And now... a brief digression on counting Sudokus.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 4!

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 4!·2·2

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 4!·2·2·3 = 288

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In contrast, there are 9x9 Sudokus. 6,670

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In contrast, there are 9x9 Sudokus. 6,670,903,752,021,072,936,960 ≈6.7x10 21

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In contrast, there are 9x9 Sudokus. 6,670,903,752,021,072,936,960 ≈6.7x10 21 (Felgenhauer-Jarvis)

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

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs gives an upper bound for the number of Sudokus of size n 2 x n 2.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs No exact formula is known, however. gives an upper bound for the number of Sudokus of size n 2 x n 2.

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

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...

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...

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.

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.

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

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

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

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

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

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

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

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.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or Note that these come from permutation matrices.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or Compute the remaining “pure” 4 th degree terms similarly.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or Even after relabelling and changing sign, the coefficients will be all positive or all negative.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or Now we compute the pure 4 th degree part of the determinant of the other mini- Sudoku.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs or But for the other one, two are positive and two are negative.

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

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.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So, there are two equivalence classes of 4 x 4 Sudokus.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So, there are two equivalence classes of 4 x 4 Sudokus. In contrast, there are equivalence classes of 9x9 Sudokus.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So, there are two equivalence classes of 4 x 4 Sudokus. In contrast, there are equivalence classes of 9x9 Sudokus. 5,472

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So, there are two equivalence classes of 4 x 4 Sudokus. In contrast, there are equivalence classes of 9x9 Sudokus. 5,472,730,538 ≈5.5 billion

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So, there are two equivalence classes of 4 x 4 Sudokus. In contrast, there are equivalence classes of 9x9 Sudokus. 5,472,730,538 ≈5.5 billion (Jarvis-Russell)

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

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Is the determinant a complete invariant for 9 x 9 Sudokus? Question: (The pure 9 th degree part of the determinant is certainly not a complete invariant, since 2 9 =512 is much less than 5.5 billion.)

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s another complete invariant for mini- Sudokus, one which is easier to compute than the determinant.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs For example, let’s determine whether these two mini- Sudokus are equivalent.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Look at the parity of the number of distinct entries along the main diagonal.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Look at the parity of the number of distinct entries along the main diagonal. 2 distinct entries

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Look at the parity of the number of distinct entries along the main diagonal. 2 distinct entries4 distinct entries

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Look at the parity of the number of distinct entries along the main diagonal. 2 distinct entries4 distinct entries Both even—same parity—so they’re equivalent.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs This method does not work for larger Sudokus. 2 distinct entries4 distinct entries Both even—same parity—so they’re equivalent.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Artpay Ootay: Udokusay anday Aphgray Oloringscay

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Artpay Ootay: Part Two: Sudokus and Graph Colorings Udokusay anday Aphgray Oloringscay

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Every mini- Sudoku can be thought of as a graph coloring, in the following way.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Think of the sixteen cells as vertices.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Connect two vertices by an edge if they are in the same row, column, or 2x2 block.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s jiggle the vertices a little so we can better see what’s happening.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs A mini- Sudoku is nothing more and nothing less than a 4-coloring of this graph.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Each vertex is either red, blue, green, or black, and no two adjacent vertices have the same color.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs One can make similar graphs for Sudokus of size...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs One can make similar graphs for Sudokus of size... 9 x 99 x 9

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs One can make similar graphs for Sudokus of size... 9 x 99 x 9 16 x 16

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs One can make similar graphs for Sudokus of size... 9 x 99 x 9 16 x 16 n2 x n2n2 x n2

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs This is the point of view in the Herzberg-Murty article. It is also the point of view of...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs It is also the point of view of

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs This is the point of view in the Herzberg-Murty article. It is also the point of view of... a Master’s Thesis in which Yato shows that solving Sudokus is an NP-complete problem.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 'ay' wey: Ghommey joq Sudokumey

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 'ay' wey: Ghommey joq Sudokumey Part Three: Groups and Sudokus

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

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

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

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

So the group of mini -Sudoku-preserving column symmetries is isomorphic to D 8, the group of symmetries of a square. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

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

Similary, one sees that the group of mini -Sudoku-preserving row symmetries is isomorphic to D 8. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs If you first do a row switch...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... then do a column switch...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... it’s the same as first doing the column switch...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... then doing the row switch.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs That is, the group R≈D 8 of permissible row permut ations commutes with the group C≈D 8 of permissible column permutations.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs That is, the group R≈D 8 of permissible row permut ations commutes with the group C≈D 8 of permissible column permutations. Moreover, a nontrivial row permutation never has the same effect as a column permutation.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In other words, the group of all combinations of row and column permutations is a direct product:

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now let’s throw the transpose into the mix.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now let’s throw the transpose into the mix. The transpose has order 2—doing it twice is the same as doing nothing at all.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now let’s throw the transpose into the mix. The transpose has order 2—doing it twice is the same as doing nothing at all. In other words, the “transpose group” is:

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs If you first switch columns...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... then take the transpose...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... it’s not the same as taking transpose, then switching columns.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Instead, it’s the same as transposing, then switching rows.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs (Taking transpose does not commute with column permutations.)

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So the group of all combinations of row and column permutations as well as transposes is not a direct product, but rather a “semidirect product.”

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We can also view this group as the automorphism group of the mini- Sudoku graph.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Finally, we wish to consider relabellings.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s a Sudoku we’ve considered before, where the entries are variables.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s number each of the cells...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... and think of the Sudoku as a function from the cells to the labels.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Every relabelling is a permutation of the labels.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Every relabelling is a permutation of the labels. E.g., here’s the relabelling that switches y and z.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So the group of relabellings is the symmetric group S 4.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs If you have a Sudoku...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs If you have a Sudoku... For the sake of sanity, only cells 1 through 8 are shown.

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

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... you can create a new Sudoku via function composition.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Row permutations, column permutations, and transposings are all cell permutations.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs For example, what does this cell permutation do?

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs It switches the top two rows.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs As before, we get new Sudokus from old ones via function composition.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s a Sudoku.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now switch the top two rows.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs I claim that cell permutations commute with relabellings.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs After all, if you’re given a Sudoku...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs After all, if you’re given a Sudoku... For the sake of sanity, only cells 1 through 8 are shown.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs For the sake of sanity, only cells 1 through 8 are shown.... it doesn’t matter if you first relabel...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs For the sake of sanity, only cells 1 through 8 are shown.... then perform a cell permutation...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... or if you first perform the cell permutation... For the sake of sanity, only cells 1 through 8 are shown.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... then relabel. For the sake of sanity, only cells 1 through 8 are shown.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Imagine for a moment that a relabelling has the same effect on every Sudoku as some cell permutation.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Imagine for a moment that a relabelling has the same effect on every Sudoku as some cell permutation. Then it commutes with every other relabelling—we just showed that.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Imagine for a moment that a relabelling has the same effect on every Sudoku as some cell permutation. Then it commutes with every other relabelling—we just showed that. So it’s in the center of S 4.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Imagine for a moment that a relabelling has the same effect on every Sudoku as some cell permutation. Then it commutes with every other relabelling—we just showed that. So it’s in the center of S 4. In which case, it equals the identity element.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Imagine for a moment that a relabelling has the same effect on every Sudoku as some cell permutation. Then it commutes with every other relabelling—we just showed that. So it’s in the center of S 4. In which case, it equals the identity element. Therefore: the set of relabellings has trivial intersection with the set of cell permutations.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So the group of all combinations of row and column permutations, transposes, and relabellings is:

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs row permutations column permutations transpose relabellings So the group of all combinations of row and column permutations, transposes, and relabellings is:

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs pt4: ddddd Sdkuz

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs pt4: Part Four: Multi-dimensional Sudokus ddddd Sdkuz

Sudokus are two-dimensional. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

Sudokus are two-dimensional. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs There is no need to limit oneself to 2D, however.

Sudokus are two-dimensional. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs There is no need to limit oneself to 2D, however. Here, for example, is a three-dimensional version of a mini- Sudoku.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In the top, we put an 8x8 grid consisting of 4 mini- Sudokus.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Below that, another one.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs And so on, for a total of 8 layers.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs The digits 1–8 must each appear once in...

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

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

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... each vertical “tower”...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs... and each 2x2x2 cube.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Is it even possible to do this at all?

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s the idea behind a quick inductive proof that the answer is yes.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s start with the top.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We already know we can make a mini- Sudoku..

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s put one in the upper left.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now add 4 to each entry, and put that next to it.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In the bottom, repeat the top, but switched around.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs That gives us the top layer.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now let’s get the next layer.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We take the top layer...

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We take the top layer... and “reverse” it.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now we’ve got the top two layers.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s get the next two layers.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs To do this, let’s go back to the original mini- Sudoku.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs But switch columns 1 and 2. Ditto for columns 3 and 4.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now apply the same procedure from before.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs That gives us the next two layers.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now let’s get the last four layers.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Again, we go back to the original mini- Sudoku.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs This time, we switch columns 1 and 3. Also 2 and 4.

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Now repeat the entire procedure, from the beginning.

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

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Idea for an REU project: use the techniques of the Herzberg-Murty paper to estimate the number of multi-dimensional Sudokus.

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.

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Presentation on theme: "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."— Presentation transcript:

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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.

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Presentation on theme: "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."— Presentation transcript:

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