Nice loops: close an alternating inference chain

In candidate-node notation, a continuous nice loop alternates strong and weak links all the way around, including the join. It does not choose one corner as true. A discontinuous loop alternates except at one candidate: two weak links meeting there rule it out; two strong links meeting there force it true. Every link must be justified.

Key idea

In a continuous loop, each weak link also gains a strong inference: at least one of its endpoints must be true. Remove candidates that conflict with both endpoints.

See the move, step by step

Trace a four-node loop for candidate 7 across rows 1 and 4.

Step 1 of 4

Choose a candidate network

Filter for candidate 7 and locate conjugate pairs that can be connected through shared units.

Rows containing the loop

Step 2 of 4

Alternate strong and weak links

The four corner candidates form a closed loop: strong across each row and weak up each column.

Loop unitsAlternating candidate loop

Step 3 of 4

Use the loop's consequence

R5C1 sees both loop states along column 1, so candidate 7 cannot remain there.

Nice loopCandidate to remove

Step 4 of 4

Keep the loop, remove the outsider

The loop itself stays unresolved while R5C1 loses candidate 7.

Nice loopUpdated target

Try it on today's puzzle

A strong link means at least one endpoint is true; a weak link means at most one is true. In this example, either R1C1 and R4C3 contain 7, or R1C3 and R4C1 contain 7. R5C1 sees a 7 in column 1 in both cases, so its 7 is removed while the corners remain unresolved. Merely returning to the starting assumption is not a contradiction; count implications, not an odd or even number of weak links.

Play today