by **Smythe Dakota** » Sun Feb 12, 2006 7:25 am

What if we consider only solved (and valid) puzzles, i.e. those with all 81 spots filled in?

I think it's reasonable to define two such puzzles to be equivalent if, and only if, one can be obtained from the other by one of the following:

1. interchanging two "minor" columns in the same "major" column, e.g. interchanging columns 1 and 2, or 1 and 3, or 2 and 3. (A "major" column consists of three columns, either 1-2-3 or 4-5-6 or 7-8-9.)

2. interchanging two major columns, e.g. 1-2-3 and 4-5-6.

3. same as 1. but with rows instead of columns.

4. same as 2. but with rows instead of columns.

5. rotating the entire puzzle 90 degrees.

6. replacing each digit in the puzzle with its image under some 1-to-1 mapping of the set {1,2,3,4,5,6,7,8,9} onto itself.

7. combining and/or repeating any of the above.

Note that reflections (horizontal, vertical, or diagonal) can be constructed from sequences of the above.

It is certain (I hope) that the above definition of equivalence is, indeed, an equivalence relation.

Is it known how many equivalence classes there are?

Bill Smythe