Almost Locked Sets
Short answer: count a group of cells in one unit against the shapes they can hold. When N cells hold exactly N+1 shapes between them, they are an almost locked set — one shape short of a naked subset. Take any single one of those shapes away from outside, and the rest lock into place. That is the whole idea, and every technique on this tier with "ALS" in its name is built on it.
What an almost locked set looks like
You already know the locked version. Three cells of a unit holding only three shapes between them are a naked triple: the three shapes are spoken for and leave the rest of the unit. Now add one more shape to the pool. Three cells, four shapes — nothing is decided, no shape is spoken for, and the subset rule says nothing at all. That is an almost locked set, and it is one shape away from being a locked one. The trick is that you do not have to wait: if something outside the set can take one of its shapes off it, the set locks on the spot.
Count the marks. The three cells in the box carry four shapes between them, so nothing is settled yet. But the plus appears in only one of those three, and that cell shares a column with the corner cell, which also holds the plus — they cannot both be it. Take the plus away from the three and they are three cells over three shapes, a locked set, and the only one of them that can be the ring is the top one. Take the plus away from the corner instead and the corner is the ring. One of those two cells is a ring either way, so every cell that sees both of them loses it.
How to use an almost locked set
- Count a unit. Take the open cells of one row, column or box and count the different shapes any group of them can hold. N cells holding N+1 shapes is an almost locked set. Start small — a cell with two marks is the smallest one there is, which is why bivalue cells are worth so much.
- Find a partner set. Look for a second almost locked set that shares at least two shapes with the first and no cell at all. The easiest partner to find is a lone bivalue cell elsewhere in the grid, and that is the pairing the board above shows.
- Check for a restricted shape. Take a shape both sets hold and ask whether every cell that could take it in one set sees every cell that could take it in the other. If so, the shape lands on at most one side — and the side that misses out is N shapes in N cells, which locks.
- Strike the other shared shape. Now take a second shape both sets hold. Whichever side locked has to use it up, so it lives inside the two sets no matter which way the first shape fell. Any cell outside both that sees every spot the shape has left in either set can therefore lose it.
The pairing above is the smallest one worth drawing, and it is also the one you will actually spot: a set you counted plus a bivalue cell you already had written down. Bigger pairings — two multi-cell sets, or a chain of them — argue exactly the same way and simply take longer to see.
Almost locked sets vs naked subsets
A naked subset is a finished argument: the cells hold as many shapes as there are cells, so those shapes belong to them and leave the rest of the unit. An almost locked set is that argument with the last step missing — it eliminates nothing on its own, and a solver who only knows subsets walks straight past it. Its value is that it is conditional: pair it with anything that can take one shape off it and it collapses into a naked subset, which is why so many hard patterns turn out to be two of these in a coat.
Find Almost Locked Sets in your own grid
Paste a puzzle and this page will look for Almost Locked Sets in it — the same pattern shown above, in your grid instead of ours. Nothing is uploaded; it runs in your browser.
Frequently asked questions
Why does a single cell count as a set?
Because the definition is N cells holding N+1 shapes, and a cell with two marks is N = 1: one cell, two shapes. It behaves exactly like the bigger ones — take one of its two shapes away and the other is forced. Treating a bivalue cell as the smallest almost locked set is not a technicality, it is what lets one rule cover the whole family.
What does "restricted" mean here?
That the shape has nowhere to hide. A shape both sets hold is restricted when every cell that could still take it in the first set sees every cell that could still take it in the second — so at most one of the two sets ends up with it. Without that condition both sets could take the shape at once and neither is forced to lock, which is the whole engine of the technique.
Is this not just an XY-Wing?
An XY-Wing is the smallest case of it: the pivot and one wing are two cells holding three shapes — an almost locked set — and the second wing is a bivalue cell, which is another. XYZ-Wing and WXYZ-Wing are the same shape of argument one size up. Learning the sets is worth it because they keep working where the wings stop: the wings need a pivot every other cell can see, and a pair of sets does not.
How big can a set get?
In principle as large as a unit. In practice form9 stops looking at four cells, the same ceiling the naked quad page stops at — a five-cell set spans most of a unit and its pencil marks are a wall nobody can read, so anything it finds you would have found another way first.
How often is it worth hunting for?
Not often, and only once the ordinary questions have all been asked — many grids that hold a pairing like this also let something simpler reach the same marks. The board above is deliberately not one of those: none of the twenty-three easier techniques form9 knows can strike the ring from either cell. If you would rather have the check done for you, the sudoku solver walks a grid one technique at a time and names each move.