Sudoku Algorithms

The following example shows a 3D Medusa chain that includes candidates 1, 2, 3, 5, and 9. The red candidate 3 appears twice in column 4 (cells C4 and H4). Candidates colored red cannot be a solution to the puzzle since they should not appear twice in the same row, column, or square. Therefore, we can remove all red candidates, leaving the yellow hints as the valid candidates.

An image on the left shows an example of the 3D Medusa Two Colours ‘Elsewhere’ technique. The chain alternates between yellow and orange. Candidate '8', marked in red in cell I2, is an 'off-chain' candidate. It can ‘see’ two chain candidates 8: the yellow candidate '8' located in the same row I in cell I8 and the orange candidate '8' located in the same 3x3 square in cell H3. The rules of simple coloring suggest that either all yellow candidates are solutions and orange candidates can be removed, or the opposite: all orange candidates are solutions and yellow candidates can be removed. In the first scenario, the yellow candidate 8 in cell I8 is a solution, and candidate 8 in cell I2 should be removed. In the second scenario, the orange candidate 8 in cell H3 is a solution, and candidate 8 in cell I8 should be removed. So, in any scenario, candidate 8 in cell I8 should be removed.

An image on the left presents an example of the 3D Medusa - Two Colours Unit + Cell technique. The chain alternates between two colors: yellow and orange. The 'off-chain' candidate '5' in F6, marked in red, can see the chain candidate of both colors: orange candidate '5' in cell F8 and yellow candidate '6' in cell F6. If orange candidates are the solution, then row F has value 5 in cell F8, and candidate 5 in cell F6 should be removed. If yellow candidates are the solution, then candidate 5 should be removed from cell F6 since candidate 6 should be in cell F6. Therefore, the 'off-chain' candidate '5' marked in red in cell F6 can be removed.

An image on the left shows an example of the 3D Medusa Cell Emptied By Color technique. The chain alternates between yellow and red. There are two 'off-chain' candidates, '6' and '2', marked in orange in cell G8. They can see the red candidate '6' in cell D8 and the red candidate '2' in cell G6. If red candidates present the puzzle solution, we should remove candidate 2 from cell G8 since there is already a value 2 in cell G6 of row G. Additionally, we should remove candidate 6 from cell G8 since value 6 is already in cell D8 of column 8. In this scenario, cell G6 becomes empty and the solution is invalid. So, candidates marked in red can be removed, leaving the yellow candidates as the valid solution.