The New Sudoku Players' Forum

[Pat]

hey, i'm sorry, Lardarse -— i guess what you really wanted was an explanation of the concept of puzzle-equivalence — perhaps this will help —

Equivalent = can be translated to each other by any combination of the following operations:

1. Permutations of the 9 digits
   
2. Transposing the matrix (that is, exchanging rows and columns)
   
3. Permuting rows within a band
4. Permuting columns within a stack
5. Permuting bands
6. Permuting stacks

Rotations and reflections are already included in the above,
because they can be expressed as combinations of these operations.

We want to find all the operations that we can make on a general Sudoku grid which preserve the property that every row, column and box contains each of the digits 1–9 exactly once.

The complete list of operations that we can perform is:

1. Relabel (i.e., permute) the 9 digits
   
2. Permute the stacks
3. Permute the bands
4. Permute the columns within a stack
5. Permute the rows within a band
   
6. Any reflection or rotation
   

Two puzzles are equivalent if:

- The puzzle is transposed (flipped along a diagonal)
- The order of 9x3 and/or 3x9 bands is permuted
- The order of columns or rows within bands are permuted

- The digits are permuted
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the permutations that leave a sudoku unchanged are:

(a) rotate the grid 90 degrees
(b) swap the top two rows
(c) swap the top two bands

(d) swap any two cell values for all cells with those values
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