by **geoff** » Tue Jun 14, 2005 1:15 pm

Dominic missed the interchange of rows and columns another factor of 2. This can be either by reflection in a diagonal or rotation by 90 degees, row shuffling turns one into the other. So 9! * 6^8 *2.

For any solution the operations in this set that preserve a particular solution must form a subgroup and thus divide that number. Further as two subgroups of size 8! * 6^6 can be found that always give different solutions the size of the subgroup for any solution is at most 648 = 3^4 * 2^3. I have previously posted a solution with this degree of symetry.

I call solutions derived by these operations similar to the base solution. The solution set is thus divided into "similarity classes" with 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 324 or 648 times 8! * 6^6 members. Classes with the lowest numbers of members are likely to be rare.

I have a feeling representation theory could be useful but I gave that up 30 years ago.

For those interested my 2 subgroups are genrated by

1) permutations of rows and columns (6^6), permutation of digits 2 to 9 (8!).

Each of these gives either a different placing of the number 1's or a different box 1.

2)Permutation of the numbers 1 to 9. (9!) Permutation of rows 456 or 789 (36), permutation of the same columns (36) switching of boxes 456 with 789 or 258 with 369 (4).

all these solutions differ in box 1 row 1 or column 1.

Using the second of these I have shown that for my very symetric solution any other operation can be undone by a combination of the operations listed.

p.s. my estimate for N=4 posted earlier also fails to be an integer.