So instead of the full 3x3 Sudoku, let's examine the smaller and more tractable 2x2 sudoku, and start with the base solution and adopt a 0-based index labeling:

0 1 2 3 <- columns a b c d <- internal labels w 0 1 2 | 3 4 x 1 3 4 | 1 2 ----+---- y 2 2 3 | 4 1 z 3 4 1 | 2 3 ^ ^ | rows internal labels |

The set of operations are listed as the following where the operations are also assigned a 'bit' and the order of operations are performed from top to bottom (or from least value to greatest). This then gives each generated solution a "value" and permutation, where the permutation is of the symbols "1234".

Operation | Bit Value | Explanation |
---|---|---|

swapRwx | 0x01 | swap rows 0 and 1 |

swapRyz | 0x02 | swap rows 2 and 3 |

swapCab | 0x04 | swap columns 0 and 1 |

swapCcd | 0x08 | swap columns 2 and 3 |

swapBR | 0x10 | swap block rows |

swapBC | 0x20 | swap block columns |

swapT | 0x40 | transpose around main diagonal |

The number of different combinations becomes:

(2 x 2 x 2) x (2 x 2 x 2) x 2 x 4!
= 2^{7} x 4! = 3072

However, we shall find that this overcounts the number of solutions that can be generated by these operations by a factor of 16, and does not generate solutions that include the Christian Baune "flip".

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