Academy
Forcing ChainForcing Chain sudoku technique
A forcing chain starts at a cell with two candidates and plays out both futures. Assume the first digit and follow the implications from link to link; then return and do the same for the second. Wherever the two branches agree — some cell ends up with the same value either way, or some candidate dies in both — that agreement is a fact, because one of the two starting assumptions must be true.
How to find a forcing chain
- Start from a bi-value cell, ideally one wired into several pairs nearby.
- Branch one: assume the first candidate and write down each forced consequence in order.
- Branch two: assume the other candidate and trace it with the same care.
- Compare the branches and keep every outcome they share. That is your deduction.
The pattern on the grid
A real position — every pencil mark computed, every red candidate a verified elimination. Toggle to see the grid after the move.
XY-Chain: (3=5)r3c4 - (5=4)r6c4 - (4=5)r6c5 - (5=3)r9c5 => r2c5<>3, r9c4<>3
534719286896000000127068400482197653315826040769003020200670000600980000908004060
Standard 81-character format (0 = empty) — paste it into your solver of choice and verify the move yourself.
Worked example
Follow the bi-value cells r3c4, r6c4, r6c5, r9c5: each link shares a candidate with the next, and both ends of the chain guard the same digit. Whichever way the first cell resolves, the implications ripple to the same conclusion — r2c5<>3, r9c4<>3. The full chain is written out in the notation line below the diagram; verify each link against the pencil marks.
When it fails
One transcription slip anywhere in a branch poisons every conclusion after it. Write the links down (r4c4=2 → r7c4≠2 → r7c4=7) rather than holding them in your head — chains fail in memory, not in logic.