Techniques / Almost locked sets

Almost locked sets: ALS-XZ, Death Blossom and Sue de Coq

A set one digit short of being locked, used as a chain node.

12 techniques in this family, introduced at tier 8. How to read a sudoku grid explains the notation used below.

What these have in common

An almost locked set is N cells holding N plus 1 candidates between them. It is one candidate away from being a naked subset, and that near miss is the useful part: remove any one of its candidates and it collapses into a locked set with all the eliminations that brings.

Why the logic holds

Take two almost locked sets that share a digit X, where every X in the first sees every X in the second. They cannot both contain X, so at least one of them loses it and becomes locked. Now take a second shared digit Z. Whichever set locked, Z is confined within the other, so any cell seeing every Z in both sets cannot hold Z. That is ALS-XZ, and chains of the same move extend it.

When to reach for it

Late, and mostly when you already suspect the grid has structure that a plain chain is not capturing. These are powerful and expensive to search by hand.

How the members differ

ALS-XZ is the two set base case. ALS Chains link more of them. ALS-XY-Wing arranges three. Death Blossom hangs an almost locked set off each candidate of a single stem cell. Sue de Coq analyses a box and line intersection as a distribution of digits that must be disjoint. Aligned Pair and Triple Exclusion come at it from the other side, ruling out combinations of two or three cells that would leave some other cell with nothing. Almost Hidden Sets are the hidden counterpart, exactly as hidden subsets are to naked ones.

The mistake to avoid

Assuming a shared digit is restricted when it is not. The whole argument depends on every instance of X in one set seeing every instance in the other. One unseen pair and the sets can both hold X, and the conclusion evaporates.

Every technique in this family

• Aligned Pair ExclusionLook at two squares together and list every pair of digits they could hold.
• Aligned Triple ExclusionLook at three squares together and list every set of three numbers they could hold.
• Almost Hidden SetThe mirror image of an almost locked set.
• Almost Locked CandidatesTwo areas cross.
• Almost Locked SetA small group of squares holding exactly one number more than there are squares.
• ALS ChainA row of almost full groups linked one to the next.
• ALS-XY-WingThree almost full groups arranged like a wing.
• ALS-XZTwo of those almost full groups share numbers.
• Death BlossomOne square is the stem.
• Subset ExclusionThe same idea for any size of group.
• Sue de CoqLook where a row or column crosses a box.
• Sue de Coq (extended)The same counting trick with more squares involved.