Simple Coloring
Simple Coloring follows a chain of cells linked in pairs for one specific digit and colors every other cell. One color must be correct, and that can be used to eliminate candidates.
Learn the techniqueAIC (Alternating Inference Chains)
AIC stands for Alternating Inference Chains and consists of chains that alternate between strong and weak links across the puzzle.
See the technique in practice
Work through the examples step by step. Each step explains what you see on the puzzle and why the conclusion holds.
Example:
- We start at row 3, column 8 with digit 3.
- A strong link leads forward to row 3, column 8. If 3 is not true in the previous cell, then 9 must be here.
- A weak link leads forward to row 3, column 9. If 9 is true in the previous cell, then 9 cannot be here.
- A strong link leads forward to row 3, column 9. If 9 is not true in the previous cell, then 3 must be here.
- The chain alternates between strong and weak links from end to end, so at least one of row 3, column 8 and row 3, column 9 must contain 3. We remove it from the notes in row 3, column 2, row 3, column 5 and row 3, column 6.
How to recognize the pattern
AIC stands for Alternating Inference Chains, chains that alternate between strong and weak links. A strong link means at least one of two candidates must be true, such as the two only placements of a digit in a unit, or the two candidates in a two-valued cell. A weak link means at most one can be true, like two identical candidates in the same unit.
When the chain alternates strong, weak, strong, weak all the way through, the ends behave like a regular strong pair: at least one of them must be true. If the chain starts and ends with the same digit, the digit can be removed from all cells that see both ends. AIC is the general framework behind many named techniques, and both coloring and XY-Wing can be written as short AIC-chains.
Step-by-step procedure
- Find a strong link for a digit and let one end be the starting point of the chain.
- Build forward by alternating: after a strong link comes a weak one, and after a weak link a new strong one must come.
- Continue until you reach a candidate with the same digit as the start, with strong links at both ends.
- Remove the digit from all cells that see both ends of the chain.
Common mistakes
- Using two weak links in a row. Then the logic breaks, and the conclusion at the end of the chain is worthless.
- Believing the ends are fact. The chain only proves that at least one of the ends is correct, not which one.
- Treating a strong link as one-way. A strong link can always be used as weak, but a weak link can never be used as strong.
When do you need the technique?
The hardest puzzles require techniques that follow long logical chains through the entire puzzle. They are in practice proof by contradiction: assume something, follow the consequences and see what does not hold. Try your way through the examples below, step by step, using the same tools the solver uses on your own puzzle.
Try it on your own puzzle
Enter your puzzle in the Sudoku Solver and it will find the next step and explain the technique behind it.
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