AIC (Alternating Inference Chains)
AIC stands for Alternating Inference Chains and consists of chains that alternate between strong and weak links across the puzzle.
Learn the techniqueForcing Chains
Forcing Chains test both candidates in a cell and follow the consequences of each of them.
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:
- The cell at row 1, column 7 has only two candidates, 1 and 2. We test both and follow the consequences.
- First assume the cell becomes 1. This assumption forces several other cells to specific digits, marked in blue.
- Now assume instead that the cell becomes 2. This assumption forces other cells to specific digits too, marked in orange.
- The assumption that the cell is 1 leads to a cell with no remaining candidates—a contradiction. Therefore, 2 must go here instead.
How to recognize the pattern
Forcing Chains take as their starting point a cell with exactly two candidates and test both. You follow the consequences of each candidate through the puzzle, placement by placement, and see where the two hypotheses end up. The technique is brute force put into a system, and it is often used when all pattern techniques have hit a wall.
Two outcomes give a valid conclusion. If both hypotheses lead to the same conclusion, such as the same cell getting the same digit, the conclusion is true regardless. And if one hypothesis leads to a contradiction, such as a cell with no candidates left, the cell must have the other candidate.
Step-by-step procedure
- Choose a cell with exactly two candidates.
- First assume one candidate and follow all forced consequences, that is, new Naked and Hidden Singles that arise.
- Repeat with the other candidate.
- If both branches lead to the same placement or same removal, that is valid regardless. If one branch leads to a contradiction, place the other candidate in the starting cell.
Common mistakes
- Mixing the branches. The consequences of the two hypotheses must be kept strictly separate, or you prove nothing.
- Following only one branch and placing the result. Without a contradiction in the other branch, one branch is just a guess.
- Following consequences that are not forced. Only placements that follow by necessity, like Singles, can be used in the chain.
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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