Almost Locked Sets

An Almost Locked Set is a group of n cells inside one unit holding exactly n + 1 different candidates — one digit more than it can use. Take any single digit away from it and it collapses into a locked set, where every remaining digit is forced to appear. Two of these sets working against each other produce eliminations no wing or fish can reach.

What the pattern is

Compare it to the sets you already know:

A single bivalue cell is the smallest Almost Locked Set: one cell, two candidates.

The working rule is called ALS-XZ. Take two Almost Locked Sets, A and B, that share at least one unit and no cells. A digit x present in both is a restricted common when every x in A sees every x in B — so x can land in one set or the other, never both. Now pick a second shared digit z. Whichever set misses out on x locks, and a locked set contains all of its digits, including z. So z sits in A or in B, and any cell outside both that sees every z in both sets cannot be z.

When to look for it

Scan for them once naked pairs and triples have stopped paying. Any two or three cells in a row, column or box with one spare digit qualify, so they are everywhere — the skill is picking pairs of sets that share a unit and testing the restricted common. Sets of two or three cells are enough for almost every position.

How to apply it, step by step

1. List small groups of cells in one unit whose combined candidates number one more than the cells.
2. Pair two such groups that share a unit and use no common cell.
3. Find a digit in both whose positions all see each other. That is your x.
4. Choose another digit z in both sets.
5. Mark every cell holding z in either set.
6. Remove z from any outside cell that sees all of those marked cells.

A worked example

A (column 1 / box 1)   r1c1 {1,2}   r2c1 {2,3}   -> 2 cells, digits {1,2,3}
B (row 1)              r1c8 {1,3}   r1c9 {3,4}   -> 2 cells, digits {1,3,4}

x = 1   r1c1 and r1c8 both lie in row 1, so only one of them can be 1
z = 3   present in both sets

If A misses the 1 it must use {2,3} and holds a 3. If B misses the 1 it must use {3,4} and holds a 3. Only one set can take the 1, so the other one carries a 3. The 3s sit at r2c1 in A and at r1c8 or r1c9 in B. Cells that see all three of those are r2c7, r2c8 and r2c9: row 2 covers r2c1, box 3 covers the other two. Those three cells lose their 3.

Common mistakes

• Counting a group as an Almost Locked Set when its cells span two units. All cells must share one row, column or box.
• Reusing a cell in both sets. They must be disjoint.
• Treating any shared digit as the restricted common without checking visibility.
• Eliminating from cells that see only part of the z candidates, or from cells inside the sets themselves.

Where you meet it

Almost Locked Sets sit in the extreme tier, alongside long chains. They earn their place in Legendary puzzles, where a single ALS-XZ often replaces a chain ten links long. They are also the raw material for Death Blossom, so learning them opens two techniques at once.