The New Sudoku Players' Forum

[Smythe Dakota]

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

[Red Ed]

Is it known how many equivalence classes there are?

Yes: 5472730538.

[gsf]

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:

the lexicographic canonicalization routine I posted takes this into account
however, instead of attempting to find a map from A=>B to show equivalence
it compares canon(A) vs. canon(B), which is more efficient for puzzle cataloging
i.e., the heavy computations are done up front once for each puzzle
then a simple 81 byte comparison (or less if per puzzle compression is used)
determines equivalence

[Smythe Dakota]

Suggestion to everyone:

Instead of the word "canonicalize", let's just use "canonize", one of the better made-up words I've ever made up.

Bill Smythe

[gsf]

Suggestion to everyone:
Instead of the word "canonicalize", let's just use "canonize", one of the better made-up words I've ever made up.

canonicalize already defined and it fits in this discussion
canonize already coined by the vatican

[Smythe Dakota]

.... canonize already coined by the vatican

That's another reason I like it.

Bill Smythe

[deam3r]

jeez:

123
456
789

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Code: Select all

1123
1234
3425
1236

EhEm?