in actual fact there are only 560,151 essentially different grids having nontrivial automorphism group

Well, nobody replied, but I worked it out, and in the process have learned some interesting facts that I will describe here.

- Code: Select all
` 5,472,730,538 number of unique Sudoku solution grids`

3,359,232 possible geometric permutations of solution grid

362,880 number of ways to renumber a grid

6,671,248,172,291,458,990,080 5,472,730,538 * 3,359,232 * 362,880

6,670,903,752,021,072,936,960 exact number of Sudoku solution grids

-----------------------------

344,420,270,386,053,120 grid deficiency

Now that last figure divided by 9! is 949,129,933,824, let's call this D.

D is exactly the number of "invariant grids" reported in the Russell & Jarvis table:

- Code: Select all
`Class Size(N) Invariants (A)`

===================================

1 1 18,383,222,420,692,992

7 96 21,233,664

8 16 107,495,424

9 192 4,204,224

10 64 2,508,084

22 5184 323,928

23 20736 162

24 20736 288

25 144 14,837,760

26 1728 2,592

27 288 5,184

28 864 2,085,120

29 3456 1,296

30 1728 294,912

31 2304 648

32 1152 6,342,480

37 1296 30,258,432

40 10368 1,854

43 93312 288

79 2916 155,492,352

86 69984 13,056

134 972 449,445,888

135 3888 27,648

142 31104 6,480

143 15552 1,728

144 15552 3,456

145 7776 13,824

===================================

T = Sum(N.A) 18,384,171,550,626,816

U = T / 3359232 5,472,730,538

===================================

The "grid deficiency" D is just the Sum of (N x A) in the table, but

excluding the value for Class 1. So D = T - A(1).

Now there is another table, produced originally by

gsf, that lists the number of essentially different grids by automorphism group:

- Code: Select all
`Aut# Grids `

----------------

1 5472170387

2 548449

3 7336

4 2826

6 1257

8 29

9 42

12 92

18 85

27 2

36 15

54 11

72 2

108 3

162 1

648 1

===============

5472730538 total

- 5472170387

560151

===============

Now I'm not certain, but believe

gsf produced this table independently, thus providing confirmation of

Red Ed's results.

And I have done a little subtraction there at the end to find 560,151 which is the number I asked about.

Finally, an interesting fact about D = 949,129,933,824, the number of individual cases of automorphism identified by Russell & Jarvis. I'd have thought that to be a "fundamental" number in the Sudoku world, just as 5472730538 is.

Remarkably, a Google search for "Sudoku 949,129,933,824" turned up just one hit! That has never happened to me before - the hit was a sample page of a book called "Mathematical Sudoku Theory" (written in German).

Will this post make that search term now turn up TWO hits?