Forcing chains: compare every branch

A forcing chain follows the consequences of a candidate choice. For a cell forcing chain, consider every candidate in one cell; for a unit forcing chain, consider every remaining position of one digit in a row, column, or box. A shared conclusion is safe only after every starting case has been checked. A contradiction instead disproves the assumption that caused it.

Key idea

If either value of the starting bivalue cell forces candidate Z into a peer of the target, Z can be removed from that target.

See the move, step by step

Use the same worked example as the XY-Wing lesson to follow both possibilities: R1C3 is either 5 or 7. The notes include earlier eliminations.

Step 1 of 4

Choose a controlled branch point

R1C3 has exactly two candidates, so the analysis has two exhaustive cases.

Branch regions

Step 2 of 4

Trace both implications

If R1C3 is 5, R1C4 is forced to 6. If R1C3 is 7, R2C3 is forced to 6.

Branch regionsTwo forcing paths

Step 3 of 4

Find the shared result

R2C4 sees both possible forced 6s, so it cannot contain 6 under either branch.

Forcing chainsCandidate to remove

Step 4 of 4

Keep only the branch-independent conclusion

Remove 6 from R2C4. Do not place any assumption candidate unless one branch was disproved.

Forcing chainsUpdated target

Try it on today's puzzle

Use short chains after simpler moves stall. An alternating inference chain (AIC) alternates strong links (if one candidate is false, the other must be true) with weak links (if one is true, the other must be false). In this grid, if R1C4 is not 6, it must be 5; R1C3 then cannot be 5, so it is 7; R2C3 then cannot be 7, so it is 6. Thus R1C4 or R2C3 must contain 6, and their shared peer R2C4 cannot. This is the same elimination written as an AIC. Check each link against the current notes, not just its drawn shape. The previous Nice Loops lesson shows alternating links in a closed loop.

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