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”) Mike Krebs, Cal State LA(joint work with C. Arcos and G. Brookfield) For slideshow: click “Research and Talks” from (Japanese for “ 数字は独身に限る ”)

For slideshow: click “Research and Talks” from 数字は独身に限る

For slideshow: click “Research and Talks” from 数字は独身に限る suji wa dokushin ni kagiru

For slideshow: click “Research and Talks” from 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 数字は独身に限る 数独

For slideshow: click “Research and Talks” from 数字は独身に限る 数独 su doku suji wa dokushin ni kagiru

For slideshow: click “Research and Talks” from 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

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

Typically, one is given some of the entries and must fill in the remaining ones. For slideshow: click “Research and Talks” from

Typically, one is given some of the entries and must fill in the remaining ones. For slideshow: click “Research and Talks” from 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 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

(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

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 colors... For slideshow: click “Research and Talks” from

The digits 1 through 9 are just labels. They could just as well be variables or colors... For slideshow: click “Research and Talks” from or...

For slideshow: click “Research and Talks” from Divizio Unu: Ekvivalenta malgranda Sudokoj

For slideshow: click “Research and Talks” from 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

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

You may not switch columns willy-nilly; tan and lavender either switch or stay fixed. 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 “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 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 So, every mini- Sudoku is equivalent to one of three mini- Sudokus.

For slideshow: click “Research and Talks” from And now... a brief digression on counting Sudokus.

For slideshow: click “Research and Talks” from 4!

For slideshow: click “Research and Talks” from 4!·2·2

For slideshow: click “Research and Talks” from 4!·2·2·3 = 288

For slideshow: click “Research and Talks” from In contrast, there are 9x9 Sudokus. 6,670

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

For slideshow: click “Research and Talks” from 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

For slideshow: click “Research and Talks” from gives an upper bound for the number of Sudokus of size n 2 x n 2.

For slideshow: click “Research and Talks” from gives an upper bound for the number of Sudokus of size n 2 x n 2.

For slideshow: click “Research and Talks” from 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 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 Note that these come from permutation matrices.

For slideshow: click “Research and Talks” from or Compute the remaining “pure” 4 th degree terms similarly.

For slideshow: click “Research and Talks” from or Even after relabelling and changing sign, the coefficients will be all positive or all negative.

For slideshow: click “Research and Talks” from 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 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 or But for the other one, two are positive and two are negative.

For slideshow: click “Research and Talks” from or Therefore, 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 So, there are two equivalence classes of 4 x 4 Sudokus.

For slideshow: click “Research and Talks” from 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 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 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 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 Is the determinant a complete invariant for 9 x 9 Sudokus? Question:

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

For slideshow: click “Research and Talks” from For example, let’s determine whether these two mini- Sudokus are equivalent.

For slideshow: click “Research and Talks” from Look at the parity of the number of distinct entries along the main diagonal.

For slideshow: click “Research and Talks” from Look at the parity of the number of distinct entries along the main diagonal.

For slideshow: click “Research and Talks” from Look at the parity of the number of distinct entries along the main diagonal. 2 distinct entries

For slideshow: click “Research and Talks” from Look at the parity of the number of distinct entries along the main diagonal. 2 distinct entries

For slideshow: click “Research and Talks” from 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 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 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 Artpay Ootay: Udokusay anday Aphgray Oloringscay

For slideshow: click “Research and Talks” from Artpay Ootay: Part Two: Sudokus and Graph Colorings Udokusay anday Aphgray Oloringscay

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

For slideshow: click “Research and Talks” from Think of the sixteen cells as vertices.

For slideshow: click “Research and Talks” from Think of the sixteen cells as vertices.

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

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

For slideshow: click “Research and Talks” from Let’s jiggle the vertices a little so we can better see what’s happening.

For slideshow: click “Research and Talks” from A mini- Sudoku is nothing more and nothing less than a 4-coloring of this graph.

For slideshow: click “Research and Talks” from A mini- Sudoku is nothing more and nothing less than a 4-coloring of this graph.

For slideshow: click “Research and Talks” from 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 One can make similar graphs for Sudokus of size...

For slideshow: click “Research and Talks” from One can make similar graphs for Sudokus of size... 9 x 99 x 9

For slideshow: click “Research and Talks” from One can make similar graphs for Sudokus of size... 9 x 99 x 9 16 x 16

For slideshow: click “Research and Talks” from 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 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 It is also the point of view of

For slideshow: click “Research and Talks” from 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 'ay' wey: Ghommey joq Sudokumey

For slideshow: click “Research and Talks” from '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

What group is it? 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 symmetries is isomorphic to D 8, the group of symmetries of a square. 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. Due to Royle.) For slideshow: click “Research and Talks” from

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

For slideshow: click “Research and Talks” from If you first do a row switch...

For slideshow: click “Research and Talks” from then do a column switch...

For slideshow: click “Research and Talks” from it’s the same as first doing the column switch...

For slideshow: click “Research and Talks” from then doing the row switch.

For slideshow: click “Research and Talks” from 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 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 In other words, the group of all combinations of row and column permutations is a direct product:

For slideshow: click “Research and Talks” from In other words, the group of all combinations of row and column permutations is a direct product:

For slideshow: click “Research and Talks” from Now let’s throw the transpose into the mix.

For slideshow: click “Research and Talks” from 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 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 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 If you first switch columns...

For slideshow: click “Research and Talks” from then take the transpose...

For slideshow: click “Research and Talks” from it’s not the same as taking transpose, then switching columns.

For slideshow: click “Research and Talks” from Instead, it’s the same as transposing, then switching rows.

For slideshow: click “Research and Talks” from (Taking transpose does not commute with column permutations.)

For slideshow: click “Research and Talks” from 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 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 We can also view this group as the automorphism group of the mini- Sudoku graph.

For slideshow: click “Research and Talks” from We can also view this group as the automorphism group of the mini- Sudoku graph.

For slideshow: click “Research and Talks” from Finally, we wish to consider relabellings.

For slideshow: click “Research and Talks” from Here’s a Sudoku we’ve considered before, where the entries are variables.

For slideshow: click “Research and Talks” from Let’s number each of the cells...

For slideshow: click “Research and Talks” from Let’s number each of the cells...

For slideshow: click “Research and Talks” from and think of the Sudoku as a function from the cells to the labels.

For slideshow: click “Research and Talks” from and think of the Sudoku as a function from the cells to the labels.

For slideshow: click “Research and Talks” from and think of the Sudoku as a function from the cells to the labels.

For slideshow: click “Research and Talks” from and think of the Sudoku as a function from the cells to the labels.

For slideshow: click “Research and Talks” from Every relabelling is a permutation of the labels.

For slideshow: click “Research and Talks” from 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 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 So the group of relabellings is the symmetric group S 4.

For slideshow: click “Research and Talks” from If you have a Sudoku...

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

For slideshow: click “Research and Talks” from and a relabelling...

For slideshow: click “Research and Talks” from you can create a new Sudoku via function composition.

For slideshow: click “Research and Talks” from Row permutations, column permutations, and transposings are all cell permutations.

For slideshow: click “Research and Talks” from For example, what does this cell permutation do?

For slideshow: click “Research and Talks” from It switches the top two rows.

For slideshow: click “Research and Talks” from As before, we get new Sudokus from old ones via function composition.

For slideshow: click “Research and Talks” from Here’s a Sudoku.

For slideshow: click “Research and Talks” from Now switch the top two rows.

For slideshow: click “Research and Talks” from I claim that cell permutations commute with relabellings.

For slideshow: click “Research and Talks” from After all, if you’re given a Sudoku...

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

For slideshow: click “Research and Talks” from 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 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 then relabel. For the sake of sanity, only cells 1 through 8 are shown.

For slideshow: click “Research and Talks” from 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 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 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 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 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 So the group of all combinations of row and column permutations, transposes, and relabellings is:

For slideshow: click “Research and Talks” from 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 pt4: ddddd Sdkuz

For slideshow: click “Research and Talks” from pt4: Part Four: Multi-dimensional Sudokus ddddd Sdkuz

Sudokus are two-dimensional. For slideshow: click “Research and Talks” from

Sudokus are two-dimensional. For slideshow: click “Research and Talks” from There is no need to limit oneself to 2D, however.

Sudokus are two-dimensional. For slideshow: click “Research and Talks” from 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 In the top, we put an 8x8 grid consisting of 4 mini- Sudokus.

For slideshow: click “Research and Talks” from In the top, we put an 8x8 grid consisting of 4 mini- Sudokus.

For slideshow: click “Research and Talks” from Below that, another one.

For slideshow: click “Research and Talks” from And so on, for a total of 8 layers.

For slideshow: click “Research and Talks” from The digits 1–8 must each appear once in...

For slideshow: click “Research and Talks” from each row...

For slideshow: click “Research and Talks” from each column...

For slideshow: click “Research and Talks” from each vertical “tower”...

For slideshow: click “Research and Talks” from and each 2x2x2 cube.

For slideshow: click “Research and Talks” from Is it even possible to do this at all?

For slideshow: click “Research and Talks” from Here’s the idea behind a quick inductive proof that the answer is yes.

For slideshow: click “Research and Talks” from Let’s start with the top.

For slideshow: click “Research and Talks” from We already know we can make a mini- Sudoku..

For slideshow: click “Research and Talks” from Let’s put one in the upper left.

For slideshow: click “Research and Talks” from Now add 4 to each entry, and put that next to it.

For slideshow: click “Research and Talks” from In the bottom, repeat the top, but switched around.

For slideshow: click “Research and Talks” from In the bottom, repeat the top, but switched around.

For slideshow: click “Research and Talks” from That gives us the top layer.

For slideshow: click “Research and Talks” from Now let’s get the next layer.

For slideshow: click “Research and Talks” from We take the top layer...

For slideshow: click “Research and Talks” from We take the top layer... and “reverse” it.

For slideshow: click “Research and Talks” from Now we’ve got the top two layers.

For slideshow: click “Research and Talks” from Let’s get the next two layers.

For slideshow: click “Research and Talks” from To do this, let’s go back to the original mini- Sudoku.

For slideshow: click “Research and Talks” from But switch columns 1 and 2. Ditto for columns 3 and 4.

For slideshow: click “Research and Talks” from Now apply the same procedure from before.

For slideshow: click “Research and Talks” from Now apply the same procedure from before.

For slideshow: click “Research and Talks” from That gives us the next two layers.

For slideshow: click “Research and Talks” from Now let’s get the last four layers.

For slideshow: click “Research and Talks” from Again, we go back to the original mini- Sudoku.

For slideshow: click “Research and Talks” from This time, we switch columns 1 and 3. Also 2 and 4.

For slideshow: click “Research and Talks” from Now repeat the entire procedure, from the beginning.

For slideshow: click “Research and Talks” from Now repeat the entire procedure, from the beginning.

For slideshow: click “Research and Talks” from Voilà

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