Two Sudoku puzzles are mathematically equivalent if one can transform the first into the second via a series of zero or more operations, where the allowed operations are:
- Relabel the 9 digits
- Swap any 2 rows in a floor
- Swap any 2 columns in a tower
- Swap any 2 floors
- Swap any 2 towers
- Swap rows with columns (i.e. matrix transpose operation)
In other words, two Sudoku puzzles are mathematically equivalent if they have the same canonical form.
If a puzzle P can be solved using a set of solving techniques, then any puzzle that is mathematically equivalent to P can also be solved using the same set of solving techniques. However, Sudoku Programs may rate two mathematically equivalent puzzles slightly differently because a different solving path is used. When a program checks for patterns one digit at a time, relabeling the digits may cause the solver to find patterns of similar nature in a different order.
A symmetrical puzzle can be mathematically equivalent to an asymmetrical puzzle. From an aesthetic point of view, these puzzles are completely different.