# Scramble

When you scramble a Sudoku puzzle, you change it to a mathematically equivalent puzzle. This scrambled version requires the same solving techniques, although some Sudoku Programs may rate it differently because the solving steps are not executed in the same order, leading to a different solving path.

The following permutations can be used to scramble a standard Sudoku:

• Relabel the 9 digits (362880 permutations)
• Swap any 2 rows in a floor (6 x 6 x 6 = 216 permutations)
• Swap any 2 columns in a tower (6 x 6 x 6 = 216 permutations)
• Swap any 2 floors (6 permutations)
• Swap any 2 towers (6 permutations)
• Swap rows and columns (2 permutations)

The total number of permutations for any Sudoku is 1218998108160.

Note that reflections and rotations can be expressed as a combination of the above operations.

## Variants

Because Sudoku Variants often have additional constraints, they cannot be scrambled in the same way as a standard Sudoku. Here are some scrambling considerations for popular Sudoku variants.

### Killer Sudoku

The cages form an irregular pattern, preventing us from most scramble operations. The following operations can still be used:

• Swap rows and columns (2 permutations)
• Relabel digits 123456789 to 987654321 (2 permutations)
• Reposition one of the floors which has no cages in common with the other 2 floors (very rare)
• Reposition one of the towers which has no cages in common with the other 2 towers (very rare)
• Reposition one of the rows in a floor which has no cages in common with any other row (very rare)
• Reposition one of the columns in a tower which has no cages in common with any other column (very rare)

It is also possible to alter Killer puzzles by systematically exchanging digits (for example, swapping all 4s and 9s) and updating the cage totals; the result will be an entirely different puzzle, however, which may well have a different difficulty-level and may or may not have a unique solution. It is also possible to combine, split or redraw the cages in an existing puzzle in order to make it easier or harder; again, one must be careful to avoid creating a puzzle with multiple solutions. Both of these quasi-scrambling techniques are frequently used by Killer aficionados to create more difficult variants of already-solved puzzles.

### Sudoku-X

The diagonals limit row and column permutations.

• Relabel the 9 digits (362880 permutations)
• Swap rows 4 & 6 (2 permutations)
• Swap columns 4 & 6 (2 permutations)
• Swap and mirror the top and bottom floor (2 permutations)
• Swap and mirror the left and right tower (2 permutations)
• Swap rows and columns (2 permutations)

### Jigsaw Sudoku

The irregular nonets prevent all row and column permutations. Only a few might be possible if the jigsaw pattern is symmetrical. Digit relabeling as usual. Swapping rows and columns and reflection is also possible.

### Windoku

The extra constraints leave room to swap rows 2 & 3, 7 & 8 and columns 2 & 3, 7 & 8. Other than that, total reflection and swapping rows and columns is possible, as well as relabeling the digits.