Counting Magic Cayley-Sudoku Tables
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1 Counting Magic Cayley-Sudoku Tables Michael Ward Western Oregon University Zassenhaus Group Theory Conference May 2015 In memory of Wolfgang Kappe 1 / 48
2 Outline Counting Magic Sudoku Tables for Z 9 [Lorch & Weld 2011] 2 / 48
3 Outline Counting Magic Sudoku Tables for Z 9 [Lorch & Weld 2011] Counting Magic Cayley-Sudoku Tables for Z 9 3 / 48
4 Outline Counting Magic Sudoku Tables for Z 9 [Lorch & Weld 2011] Counting Magic Cayley-Sudoku Tables for Z 9 Counting Magic Cayley-Sudoku Tables for Z 3 Z 3 4 / 48
5 Outline Counting Magic Sudoku Tables for Z 9 [Lorch & Weld 2011] Counting Magic Cayley-Sudoku Tables for Z 9 Counting Magic Cayley-Sudoku Tables for Z 3 Z 3 Mutually Orthogonal Sets of Magic Sudoku Tables for Z 9 [Lorch & Weld 2011] 5 / 48
6 Outline Counting Magic Sudoku Tables for Z 9 [Lorch & Weld 2011] Counting Magic Cayley-Sudoku Tables for Z 9 Counting Magic Cayley-Sudoku Tables for Z 3 Z 3 Mutually Orthogonal Sets of Magic Sudoku Tables for Z 9 [Lorch & Weld 2011] Mutually Orthogonal Sets of Magic Cayley-Sudoku Tables for Z 3 Z 3 6 / 48
7 Definitions A Sudoku Table for Z 9 = {0,1,2,3,4,5,6,7,8} is a 9 9 array partitioned into 3 3 blocks in which the elements of Z 9 appear exactly once in each row and column (Latin square) and in each block (sudoku puzzle). Such a table is Magic if the row, column, and diagonal sums in each 3 3 block are zero mod 9. 7 / 48
8 Magic Sudoku Table for Z 9 Example / 48
9 Sales Pitch & Question 9 9 Latin Square Count The exact number of Latin squares of order 9 (approximately ) wasn t known until 1975, and the exact number for orders twelve and larger is currently unknown. [Lorch & Weld] 9 / 48
10 Sales Pitch & Question 9 9 Latin Square Count The exact number of Latin squares of order 9 (approximately ) wasn t known until 1975, and the exact number for orders twelve and larger is currently unknown. [Lorch & Weld] Sudoku Table Count The exact number of Sudoku Tables for Z 9 is approximately , evidently known only via computer. 10 / 48
11 Sales Pitch & Question 9 9 Latin Square Count The exact number of Latin squares of order 9 (approximately ) wasn t known until 1975, and the exact number for orders twelve and larger is currently unknown. [Lorch & Weld] Sudoku Table Count The exact number of Sudoku Tables for Z 9 is approximately , evidently known only via computer. Question How many Magic Sudoku Tables are there for Z 9? 11 / 48
12 Lorch & Weld Counting Let M be the set of Magic Sudoku Tables for Z / 48
13 Lorch & Weld Counting Let M be the set of Magic Sudoku Tables for Z 9. Find a group acting on M with few orbits. Add up the orbit lengths. 13 / 48
14 Lorch & Weld Counting Let M be the set of Magic Sudoku Tables for Z 9. Find a group acting on M with few orbits. Add up the orbit lengths. L & W construct a group of order with 2 orbits. 14 / 48
15 Lorch & Weld Counting Let M be the set of Magic Sudoku Tables for Z 9. Find a group acting on M with few orbits. Add up the orbit lengths. L & W construct a group of order with 2 orbits. Stabilizers have orders and / 48
16 Lorch & Weld Counting Let M be the set of Magic Sudoku Tables for Z 9. Find a group acting on M with few orbits. Add up the orbit lengths. L & W construct a group of order with 2 orbits. Stabilizers have orders and 2 3. M = = 32, / 48
17 Definitions A Cayley-Sudoku Table for Z 9 [C-S Table] is a Cayley table which is also a (bordered) Sudoku table Such a table is Magic if the row, column, and diagonal sums in each 3 3 block is zero mod / 48
18 Counting How many Magic Cayley-Sudoku Tables [MC-S Tables] for Z 9? 18 / 48
19 Counting How many Magic Cayley-Sudoku Tables [MC-S Tables] for Z 9? Lemma None! 19 / 48
20 Counting How many Magic Cayley-Sudoku Tables [MC-S Tables] for Z 9? Lemma None! Switch groups to Z 3 Z 3 := Z / 48
21 Constructing Magic Cayley-Sudoku Tables for Z 3 Z 3 Choose any two complementary subgroups of order 3, U =< u > and V =< v >, U V = Z 3 Z 3 (multiplicative notation). Large columns" indexed by cosets of U. Columns within each large column labeled with elements of its coset. Large rows and rows similarly labeled using cosets of V. U vu v 2 U 1 u u 2 v vu vu 2 v 2 v 2 u v 2 u u u 2 v vu vu 2 v 2 v 2 u v 2 u 2 V v v vu vu 2 v 2 v 2 u v 2 u 2 1 u u 2 v 2 v 2 v 2 u v 2 u 2 1 u u 2 v vu vu 2 u u u 2 1 vu vu 2 v v 2 u v 2 u 2 v 2 V u vu vu vu 2 v v 2 u v 2 u 2 v 2 u u 2 1 v 2 u v 2 u v 2 u 2 v 2 u u 2 1 vu vu 2 v u 2 u 2 1 u vu 2 v vu v 2 u 2 v 2 v 2 u V u 2 vu 2 vu 2 v vu v 2 u 2 v 2 v 2 u u 2 1 u v 2 u 2 v 2 u 2 v 2 v 2 u u 2 1 u vu 2 v vu 21 / 48
22 Characterization Any permutation of the large columns [rows] yields another MC-S Table. Within any large column [row], any permutation of the columns [rows] yields another MC-S Table. 22 / 48
23 Characterization Any permutation of the large columns [rows] yields another MC-S Table. Within any large column [row], any permutation of the columns [rows] yields another MC-S Table. Theorem Every MC-S Table for Z 2 3 is obtained in this way for some choice of U and V. 23 / 48
24 Characterization Any permutation of the large columns [rows] yields another MC-S Table. Within any large column [row], any permutation of the columns [rows] yields another MC-S Table. Theorem Every MC-S Table for Z 2 3 is obtained in this way for some choice of U and V. Any of the MC-S Tables so constructed from a given U and V we call a (U,V )-table. 24 / 48
25 Plain Counting 4 Choices for U =< u > 3 Choices for V =< v > 3! Arrangements of large columns (indexed by U, vu, v 2 U ) (3!) 3 Arrangements of columns within large columns 3! Arrangements of large rows (indexed by V, V u, V u 2 ) (3!) 3 Arrangements of rows within large rows 12 (3!) 8 Magic Cayley-Sudoku Tables for Z (3!) 8 = 20,155, / 48
26 Fancy Counting à la Lorch & Weld M := the set of MC-S Tables for for Z / 48
27 Fancy Counting à la Lorch & Weld M := the set of MC-S Tables for for Z 2 3 C := group of permutations of large columns and columns within large columns 27 / 48
28 Fancy Counting à la Lorch & Weld M := the set of MC-S Tables for for Z 2 3 C := group of permutations of large columns and columns within large columns C = S 3 S 3 28 / 48
29 Fancy Counting à la Lorch & Weld M := the set of MC-S Tables for for Z 2 3 C := group of permutations of large columns and columns within large columns C = S 3 S 3 R := group of permutations of large rows and rows within large rows 29 / 48
30 Fancy Counting à la Lorch & Weld M := the set of MC-S Tables for for Z 2 3 C := group of permutations of large columns and columns within large columns C = S 3 S 3 R := group of permutations of large rows and rows within large rows R = S 3 S 3 30 / 48
31 Fancy Counting à la Lorch & Weld M := the set of MC-S Tables for for Z 2 3 C := group of permutations of large columns and columns within large columns C = S 3 S 3 R := group of permutations of large rows and rows within large rows R = S 3 S 3 L := the group of relabelings, i.e. bijections from Z 2 3 to itself that preserve MC-S Tables 31 / 48
32 Fancy Counting à la Lorch & Weld M := the set of MC-S Tables for for Z 2 3 C := group of permutations of large columns and columns within large columns C = S 3 S 3 R := group of permutations of large rows and rows within large rows R = S 3 S 3 L := the group of relabelings, i.e. bijections from Z 2 3 to itself that preserve MC-S Tables L = Aut(Z 2 3 ) = GL(2,3) 32 / 48
33 Fancy Counting à la Lorch & Weld M := the set of MC-S Tables for for Z 2 3 C := group of permutations of large columns and columns within large columns C = S 3 S 3 R := group of permutations of large rows and rows within large rows R = S 3 S 3 L := the group of relabelings, i.e. bijections from Z 2 3 to itself that preserve MC-S Tables L = Aut(Z 2 3 ) = GL(2,3) Set G = L C R = GL(2,3) (S 3 S 3 ) (S 3 S 3 ) 33 / 48
34 Fancy Counting à la Lorch & Weld M := the set of MC-S Tables for for Z 2 3 C := group of permutations of large columns and columns within large columns C = S 3 S 3 R := group of permutations of large rows and rows within large rows R = S 3 S 3 L := the group of relabelings, i.e. bijections from Z 2 3 to itself that preserve MC-S Tables L = Aut(Z 2 3 ) = GL(2,3) Set G = L C R = GL(2,3) (S 3 S 3 ) (S 3 S 3 ) G = 48 (3!) 8 34 / 48
35 Fancy Counting Continued Lemma G acts transitively on M and G M = 4 for any M M. 35 / 48
36 Fancy Counting Continued Lemma G acts transitively on M and G M = 4 for any M M. Proof. Let M i M, M i a (U i,v i )-table for i = 12. For some σ GL(2,3), U σ 1 = U 2 and V σ 1 = V 2 M σ 1 is a (U 2,V 2 )-table. Permute rows and columns with an element of C R to match M / 48
37 Fancy Counting Continued Lemma G acts transitively on M and G M = 4 for any M M. Proof. Let M i M, M i a (U i,v i )-table for i = 12. For some σ GL(2,3), U σ 1 = U 2 and V σ 1 = V 2 M σ 1 is a (U 2,V 2 )-table. Permute rows and columns with an element of C R to match M 2. Fix M M, M i a (U,V )-table, U =< u >, V =< v >. σ GL(2,3), ρ C R, σρ G M U σ = U, v σ = V u σ = u ±1, v σ = v ±1 For each such σ there is a unique ρ such that σρ fixes M. 37 / 48
38 Fancy Counting Continued Lemma G acts transitively on M and G M = 4 for any M M. Proof. Let M i M, M i a (U i,v i )-table for i = 12. For some σ GL(2,3), U σ 1 = U 2 and V σ 1 = V 2 M σ 1 is a (U 2,V 2 )-table. Permute rows and columns with an element of C R to match M 2. Fix M M, M i a (U,V )-table, U =< u >, V =< v >. σ GL(2,3), ρ C R, σρ G M U σ = U, v σ = V u σ = u ±1, v σ = v ±1 For each such σ there is a unique ρ such that σρ fixes M. Corollary M = G 48 (3!)8 = = 12 (3!) 8 = 20,155,393. G M 4 38 / 48
39 Orthogonality Definition Two Latin squares are orthogonal provided each ordered pair of symbols occurs exactly once when the squares are superimposed. A family of Latin squares is mutually orthogonal provided each pair of distinct elements are orthogonal. 39 / 48
40 Orthogonality Definition Two Latin squares are orthogonal provided each ordered pair of symbols occurs exactly once when the squares are superimposed. A family of Latin squares is mutually orthogonal provided each pair of distinct elements are orthogonal. Example These Cayley tables of Z 3 are orthogonal / 48
41 Sales Pitch & Questions The next Fermat problem? Regarding families of mutually orthogonal latin squares, it has long been known that there are at most n 1 mutually orthogonal Latin squares of order n [i.e. n n], and that this bound is achieved when n is a prime power. However, for non-prime power orders larger than six, the largest size of a family of mutually orthogonal latin squares is unknown. This problem has been proposed by Mullen as a candidate for the next Fermat problem. " [Lorch & Weld] 41 / 48
42 Sales Pitch & Questions The next Fermat problem? Regarding families of mutually orthogonal latin squares, it has long been known that there are at most n 1 mutually orthogonal Latin squares of order n [i.e. n n], and that this bound is achieved when n is a prime power. However, for non-prime power orders larger than six, the largest size of a family of mutually orthogonal latin squares is unknown. This problem has been proposed by Mullen as a candidate for the next Fermat problem. " [Lorch & Weld] So the largest size of a mutually orthogonal family of 9 9 Latin squares is / 48
43 Sales Pitch & Questions The next Fermat problem? Regarding families of mutually orthogonal latin squares, it has long been known that there are at most n 1 mutually orthogonal Latin squares of order n [i.e. n n], and that this bound is achieved when n is a prime power. However, for non-prime power orders larger than six, the largest size of a family of mutually orthogonal latin squares is unknown. This problem has been proposed by Mullen as a candidate for the next Fermat problem. " [Lorch & Weld] So the largest size of a mutually orthogonal family of 9 9 Latin squares is 8. Questions What is the largest size of a mutually orthogonal family of Sudoku Tables & Magic Sudoku Tables for Z 9? Magic Cayley-Sudoku Tables for Z 2 3? 43 / 48
44 Mutually Orthogonal Sets of Sudoku Tables for Z 9 Theorem (Pedersen & Vis, 2009) There exists a family of 6 mutually orthogonal Sudoku Tables for Z 9 and this is the largest possible such family. O Orthogonal Sudoku Puzzle (MAA FOCUS) 44 / 48
45 Mutually Orthogonal Sets of Magic Sudoku Tables for Z 9 Theorem (Lorch & Weld) There exists a family of 2 mutually orthogonal Magic Sudoku Tables for Z 9 and this is the largest possible such family. 45 / 48
46 Mutually Orthogonal Sets of Magic-Cayley Sudoku Tables for Z 2 3 Theorem There exists a family of 6 mutually orthogonal Magic Cayley-Sudoku Tables for Z 9 and this is the largest possible such family, assuming any such family can be normalized to have the same column labels. 46 / 48
47 Mutually Orthogonal Sets of Magic-Cayley Sudoku Tables for Z 2 3 Theorem There exists a family of 6 mutually orthogonal Magic Cayley-Sudoku Tables for Z 9 and this is the largest possible such family, assuming any such family can be normalized to have the same column labels. Idea of the Proof (adapted from Pedersen & Vis) Think of Z 2 3 as the additive group of GF(9). Take U = GF(3). For each x GF(9)\U, Ux is a complement to U. A family of (U,Ux)-tables, one for each x GF(9)\U, suitably arranged, is mutually orthogonal. 47 / 48
48 References 1. J. Lorch and E. Weld, Modular Magic Sudoku, 2. G. L. Mullen, A candidate for the next Fermat problem," Math. Intelligencer R. M. Pedersen and T. L. Vis, Sets of Mutually Orthogonal Sudoku Latin Squares, College Math. J. 40 (2009) / 48
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