It is well known that two Sudokus (either completed grids or partially filled-in puzzles) can be considered equivalent if one grid can be obtained from the other by a series of zero or more of the following maneuvers:

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A. Interchange any two complete rows within the same band.

B. Interchange any two complete columns within the same stack.

C. Interchange any two complete bands.

D. Interchange any two complete stacks.

E. Rotate the puzzle 90 degrees.

F. Replace each digit with its image under some one-to-one correspondence of the set {1,2,3,4,5,6,7,8,9} onto itself.

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Any other type of equivalence (such as horizontal, vertical, or diagonal reflection) can be achieved with a combination of the above.

Now let's step down a bit and look at Latin squares. (A 9x9 Latin square is just a Sudoku without the boxes.) Two 9x9 Latin squares (either completed grids or partially filled-in puzzles) can be considered equivalent if one grid can be obtained from the other by a series of zero or more of the following maneuvers:

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N. Interchange any two complete rows.

O. Interchange any two complete columns.

P. Rotate the puzzle 90 degrees.

Q. Replace each digit with its image under some one-to-one correspondence of the set {1,2,3,4,5,6,7,8,9} onto itself.

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But now I propose two more (for Latin squares):

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R. Interchange rows with digits.

S. Interchange columns with digits.

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By interchanging columns with digits, for example, I mean that if rXcY contains the digit Z in the first puzzle, then rXcZ would contain the digit Y in the second puzzle.

To see that such interchanges indeed result in equivalent Latin squares, consider the set BIGTHING of all triples (X,Y,Z) where each X,Y,Z is a digit 1 through 9. There are 729 (9x9x9) elements in this set. A valid Latin square is a subset LITTLETHING of BIGTHING such that:

1. No two distinct elements of LITTLETHING have the same first and second coordinates, and

2. No two distinct elements of LITTLETHING have the same first and third coordinates, and

3. No two distinct elements of LITTLETHING have the same second and third coordinates.

If LITTLETHING has 81 elements, it represents a completed Latin square grid. If it has fewer than 81 elements, it represents a partially filled-in puzzle.

If you consider the first coordinate to represent the row, the second to represent the column, and the third to represent the digit, then the above conditions are equivalent, respectively, to:

1. There can be at most one digit in each cell.

2. No two cells in the same row can have the same digit.

3. No two cells in the same column can have the same digit.

Since the set-of-ordered-triples definition of Latin square is symmetric in its handling of the three coordinates, it follows that any two of them (e.g. columns and digits) can be interchanged, and the result will still be a valid Latin square.

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Now that I've convinced everybody, let me ask the big question:

Can something like this be done with full Sudokus (recognizing the boxes also)? i.e. is there some way of defining row-digit and column-digit interchanges that can create a new type of equivalence for Sudokus, or does it work only for Latin squares?

Bill Smythe