ALS-XZ
Short answer: find two almost locked sets with no cell in common that share two shapes, call them X and Z. If every cell that could still be X in one set sees every cell that could still be X in the other, then X lands in at most one of them — and the set that misses out locks. Z is therefore used up inside the two sets, and comes off every cell outside them that sees all of Z's remaining spots in both.
What an ALS-XZ looks like
The almost locked set page paired one set with a single bivalue cell, because that is the smallest partner there is. Nothing in the argument needed the partner to be small, though. Two full sets work the same way, and there are far more of them: the whole rule is that one shared shape can only go to one side, which leaves the other side with exactly as many shapes as it has cells. The letters are just labels — X is the shape that decides which set locks, Z is the shape you get to strike.
Work it through both ways. If the left group takes the ring, the right group is two cells holding only the triangle and the diamond — one of them has to be the triangle. If the right group takes the ring instead, the left group is two cells holding only the triangle and the diamond, and its lower cell is the triangle. The two groups cannot both take the ring, because the only cell in each that still has one sits in the same row as the other. So one of the two groups is locked, whichever it is, and in both cases the triangle is spent inside the four cells. The cell between the right-hand pair sees all three of the spots the triangle could still be using, so it cannot be the triangle itself.
How to use ALS-XZ
- Find two sets that share two shapes. Count cells against shapes in a row, column or box until you have two almost locked sets — N cells holding N+1 shapes — that share no cell at all. They need at least two shapes in common. A bivalue cell counts as a set of one, so start with the pairs you have already written down.
- Test one shared shape for restriction. Pick one shared shape and call it X. List every cell of the first set that could still be X, and every cell of the second. If every cell on one list sees every cell on the other, X is restricted: it can land in at most one of the two sets.
- Name the set that locks. Whichever set does not get X is N cells holding N shapes, which is a locked set — every shape it has left goes to one of its own cells. You do not need to know which set that is. The argument only needs it to be one of the two.
- Strike the other shared shape. Now take a second shared shape, Z. Whichever set locked has to spend its Z inside itself, so Z lives somewhere in the two sets no matter which way X fell. Any cell outside both that sees every remaining Z in the first set and every remaining Z in the second therefore cannot be Z.
Step two is the one that fails most often, so test it first and abandon the pairing quickly. Two sets sharing three shapes are worth a longer look: each shared shape is a separate candidate for X, and a pairing that fails on one may still be restricted on another.
Singly linked and doubly linked
Everything above is the singly linked rule: one restricted shape, one strike to look for. If a second shared shape turns out to be restricted as well, the pairing is doubly linked, and it gets much stronger — with two shapes each confined to one side, every shape the two sets hold between them is used up inside the pair, so each of them can be struck from the cells outside that see all of it, and not just the one you picked as Z. The doubly linked form is also the step that turns this into ALS chains, where a whole run of sets is threaded together by one restricted shape after another — see ALS chains. Learn the singly linked version first — the extra link changes what you can strike, not why it works.
Find ALS-XZ in your own grid
Paste a puzzle and this page will look for ALS-XZ in it — the same pattern shown above, in your grid instead of ours. Nothing is uploaded; it runs in your browser.
Frequently asked questions
What do X and Z stand for?
Nothing, and that is worth knowing early. They are just labels for the two shapes the sets have in common: X is the one that is restricted, so it decides which set locks, and Z is the one you go on to strike. Some solvers write the technique "ALS-XZ rule" for exactly that reason. If a pairing shares three shapes, any of them can play either part, so it is worth trying each one as X.
Why must the two sets be disjoint?
Because a cell belonging to both sets would have to see itself for the restriction to mean anything, and the counting breaks too: a shared cell is counted once in each set, so "N cells holding N+1 shapes" stops describing the pair. Overlapping sets are not a pairing at all — they are one bigger group, which may or may not be an almost locked set in its own right.
How is this different from the almost locked set page?
It is the same rule with a bigger partner. The primer pairs a set with one bivalue cell, which is the easiest pairing to spot and the one worth learning on. Here both sides are real sets, which is where the technique earns its keep: bivalue partners are scarce on a hard grid, and multi-cell ones are everywhere once you start counting.
Is an XY-Wing one of these?
Yes, and so is most of the wing family. An XY-Wing is a two-cell set — the pivot and one wing — paired with a bivalue cell, with the pivot's shared shape as X. XYZ-Wing and WXYZ-Wing are the same shape of argument with a bigger first set. The wings are worth learning as patterns because they are quick to recognise; ALS-XZ is worth learning because it keeps finding things after the wings run out.
How often is it worth hunting for?
Late, and with a plan — pick a bivalue cell or a two-cell set you already noticed, then look for a partner rather than scanning the grid for pairings at large. On the board above none of the twenty-five easier techniques form9 knows can strike the triangle from that cell, which is the bar every figure on this tier has to clear, but plenty of grids that hold a pairing let something simpler reach the same mark. The sudoku solver will do the checking for you and name whichever move comes first.