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, which is what almost locked set Sudoku patterns are for.
What the pattern is
Compare it to the sets you already know:
| Set | Cells | Candidates | Behaviour |
|---|---|---|---|
| Locked set (naked pair, triple, quad) | n | n | every digit is used, all of them leave the rest of the unit |
| Almost Locked Set | n | n + 1 | one digit too many; remove any one digit and it locks |
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 have no cell in common. They do not have to share a unit or lie in the same box; what matters is the digit that links them. 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 have no common cell and testing the restricted common. Sets of two or three cells are enough for almost every position.
How to apply it, step by step
- List small groups of cells in one unit whose combined candidates number one more than the cells.
- Pair two such groups that use no common cell.
- Find a digit in both whose positions all see each other. That is your
x. - Choose another digit
zin both sets. - Mark every cell holding
zin either set. - Remove
zfrom any outside cell that sees all of those marked cells.
A worked example
Set A is A4 plus C6 in box 2. Set B is A7 plus B9 in box 3. Both hold the digits 3, 7 and 9, and they sit in different boxes. What links them is 9: A can hold it only in A4, B only in A7, and both cells are in row A, so 9 is the restricted common digit. The second shared digit is z = 7.
If A has no 9, it is and the 7 must be in C6. If B has no 9, it is and one of A7 and B9 is 7. One of the two cases is true, so a 7 sits in C6, A7 or B9. C8 sees C6 through row C and both A7 and B9 through box 3, so C8 loses its 7.
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
zcandidates, 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.