Almost Locked Sets

What Is an Almost Locked Set?

An ALS is N cells in a single house with exactly N+1 distinct candidates. It is "almost locked" -- remove one candidate and it becomes a naked subset. The simplest ALS is a bivalue cell (1 cell, 2 candidates). All cells must share a house.

What Is a Restricted Common Candidate?

The RCC is a digit appearing in both ALS A and ALS B where every instance in A sees every instance in B. This means the digit can only go in one of the two ALSs, causing the other to become locked. A second shared digit Z can then be eliminated from cells seeing all its instances in both ALSs.

ALS-XY: The Original ALS Interaction Rule

Two ALSs share an RCC (X) and another common digit (Z). X can only go in one ALS, locking the other. Z must end up in at least one ALS. Eliminate Z from cells seeing all Z's in both. Level 9 (Master).

ALS-XZ: The Refined Two-ALS Technique

Same logic as ALS-XY with stricter validation. The algorithm enumerates all ALSs and checks every pair for RCCs. Level 10 (Master).

ALS Chain: Connecting Multiple Almost Locked Sets

A sequence of ALSs linked by RCCs (each consecutive pair using a different RCC digit). The elimination target appears in both the first and last ALSs. Level 11 (Extreme). Subsumes two-ALS techniques as special cases.

Sue de Coq: The ALS Intersection Pattern

Exploits the intersection of a block with a row or column. Two helper sets (one from the row, one from the block) with disjoint candidates that together cover the intersection's candidates. Level 9 (Master).

Death Blossom: The Stem-and-Petal ALS Technique

A stem cell with N candidates connects to N petal ALSs, one per candidate. Each candidate is the RCC with its petal. A digit appearing in all petals can be eliminated from cells seeing all its instances across petals. Level 11 (Extreme).

How to Find ALS Patterns

Catalog ALSs in each house. Look for pairs sharing candidates with valid RCCs. For Sue de Coq, focus on block-line intersections. For Death Blossom, start from stem cells with few candidates.

Difficulty Ratings

ALS-XY: Level 9, Master Sue de Coq: Level 9, Master ALS-XZ: Level 10, Master ALS Chain: Level 11, Extreme Death Blossom: Level 11, Extreme

Summary

ALS techniques build on two concepts: the Almost Locked Set (N cells, N+1 candidates) and the Restricted Common Candidate. Five techniques apply this in different configurations, forming one of the most powerful families for tackling the hardest puzzles.