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Sets as nodes in forcing chains

If a set of positions is an 'almost locked set', that is, if it has one more candidate than its size, then it can be used as node in a forcing chain: a predecessor can have a value that eliminates one of the candidates, and the resulting locked set can force exclusions.


After a bit of work, the puzzle shown is brought to this state here. Now the solution is the forcing chain (5,4)!5 > (9,4)5 > (8,569)179 > (7,4)!7 > (5,4)7 that shows that (5,4) must have 5 or 7, eliminating the other three candidates.

Here the notation (8,569)179 means that the three positions (8,5), (8,6), (8,9) contain the three values 1, 7, 9 in some unspecified order.

Afterwards things are straightforward: we find a 4 in box 5, a 4 in box 4, a double pair 56 in column 1, and then singles suffice.

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