Sudoku Forcing Chains & Alternating Inference Chains (AIC)

When geometric techniques like X-Wings, Swordfish, and Jellyfish are exhausted, Very Difficult Sudoku puzzles demand the ultimate weapon of mathematical logic: Forcing Chains and Alternating Inference Chains (AIC). Here is how grandmasters resolve seemingly impossible grids with 100% deterministic certainty.

1. What is a Forcing Chain?

A Forcing Chain tests a hypothesis on a bi-value cell (a cell with strictly two candidate numbers, such as [3, 7]). You trace the logical consequence of both possibilities:

Hypothesis A: Assume Cell X is 3 → traces through candidates → forces Cell Y to be 9.

Hypothesis B: Assume Cell X is 7 → traces through candidates → ALSO forces Cell Y to be 9!

Because Cell X must mathematically be either 3 or 7, and both possibilities force Cell Y to become 9, Cell Y MUST be 9! The deduction is complete without any guesswork.

2. The Foundation of AIC: Strong vs. Weak Links

Alternating Inference Chains connect cells using two types of logical links:

Strong Link (Double Implication)

"If NOT A, then B"

Occurs when a row, column, or block has only TWO candidates for a digit. If one is false, the other MUST be true.

Weak Link (Single Implication)

"If A, then NOT B"

Occurs when two candidates see each other in the same unit. If one is true, the other cannot be true.

3. Cell Forcing vs. Digit Forcing Chains

Cell Forcing Chain

You pick a cell with two candidates (e.g. [2, 8]). Follow all ramifications of 2 and all ramifications of 8 until they lead to the same confirmed cell value elsewhere.

Digit Forcing Chain

You pick a specific digit (e.g. 5) that only appears twice in a row or column. Follow both positions until both prove that another candidate 5 can be safely eliminated.

Ready to Practice on Very Difficult Grids?

Test Forcing Chains on our Evil and Extreme difficulty tiers with instant smart hints.

Play Evil Sudoku →