An almost locked set (ALS) is N cells in one row, column or box that hold N + 1 candidates. When two such sets share a digit that can go in at most one of them, the restricted common candidate, any other digit both hold goes from every cell that sees all its copies in both sets. The move is also called ALS-XZ.
What an almost locked set is
A naked pair is locked. Row six, columns eight and nine hold only 7 and 9, so they take the 7 and the 9. An almost locked set has one candidate too many: N cells in one house with N + 1 candidates. Row two, columns six and seven hold 2 and 9, and 2, 3 and 9. That is three digits for two cells.
Why the spare digit matters
Rule out one of its digits and the set locks. If those two cells can't hold the 3, they hold the 2 and the 9, a naked pair. Everything below uses that fact.
How to spot one
Count cells and candidates in one house. Look for two cells with three digits between them. A single cell with two candidates is the smallest almost locked set.
The restricted common candidate
Take two almost locked sets with no cell in common. A digit both hold is a restricted common candidate when every copy of it in one set sees every copy in the other. It can then go in one of the sets at most.
The second set here is column nine, rows two and three, with 3, 4 and 9. Both sets hold a 3 and a 9. Each has one 3, in row two, columns seven and nine, so the 3 is restricted.
How to spot it in play
With full pencil marks and no single left, look for two small sets close together. If every copy of a shared digit, in both sets, sits in one row, column or box, it is restricted.
Find a set with one spare digit
Row two, columns six and seven hold 2, 3 and 9 between them. Two cells, three digits.
Find a second set
Column nine, rows two and three hold 3, 4 and 9. This set shares no cell with the first.
Find the digit they can't share
Each set has one 3, and both sit in row two. They can't both be the 3, so at most one set gets it.
One set locks and holds a 9
At least one set goes without the 3 and locks. The first would hold 2 and 9, the second 4 and 9. Either way one of the sets holds a 9.
Find the cell that sees every 9
Row two, column eight sees all four 9s: three along row two and one through box three. No other cell with a 9 does.
Strike the 9
Whichever set holds a 9, row two, column eight sees it, so its own 9 goes. Only the 6 is left, so the cell is the 6.
Why it works
The restricted 3 can't go in both sets, so one set goes without it and locks. Both locked versions contain a 9, so a cell that sees all four 9s can't be the 9.
In general: two sets share the digits X and Z, and X is restricted. Any cell outside the sets that sees every Z in both sets loses its Z, hence the name ALS-XZ.
Each set's only 3 is in row two, so one set at most holds it.
Row two, column six and row three, column nine share no house. Both could be 9, so the 9 is not restricted.
How it relates to the Y-wing
A Y-wing is the smallest case. One wing alone is a one-cell set. The pivot and the other wing share a house, so together they form a two-cell set with three candidates. On the Y-wing article's grid, the game's check for this move strikes that Y-wing's 5 with just such a pair.
Here both sets have two cells. The game's Y-wing check does not remove this 9, and neither do the other wings and chains it tries first.
How it relates to naked subsets
An almost locked set is a naked subset, such as a naked triple, with one cell missing. Alone it removes nothing. A second set and a restricted digit tell it which digit is left out.
What the in-game hint shows
The in-game hint may show a different or simpler step first. On this grid it opens with a naked pair: row three, columns three and nine hold only 4 and 9, so the rest of row three loses its 9s. Both sets keep their candidates.
An XY-chain through eight cells follows, then singles. The fourth single makes row two, column eight the 6.
On the starting grid, no step the hint checks before Nishio strikes that 9 on its own. Nishio does. If row two, column eight is the 9, row two, column six is the 2 and row three, column nine the 4, and then row two, columns seven and nine both need the 3. The game checks Nishio before this move, so the hint seldom shows it. A different route can still be right.
Find two almost locked sets. If a digit they share can go in at most one of them, any other shared digit goes from every cell that sees all its copies in both sets.
Where this sits on the path
This is the second technique of the last chapter, after BUG+1. Almost locked sets return later in the chapter, in ALS chains and death blossoms. Aligned pair exclusion comes next.
On a different grid, two almost locked sets are outlined: row four, columns eight and nine, and column nine, rows one and two. Which candidate do these two sets remove?
Both sets hold a 1 and a 9. Check which of the two has all its copies in one column.
Recap
- Look for
- Two almost locked sets, N cells with N + 1 candidates each, that share two digits.
- Scope
- Two sets with no cell in common, and the cells that see them.
- Result
- The shared digit that is not restricted goes from every cell that sees all its copies in both sets.
- Tell-tale
- A shared digit whose copies in both sets all sit in one row, column or box.
- If it fails
- If no shared digit is restricted, try another pair of sets.