ALS Chains
Short answer: line up three almost locked sets so that each one shares a restricted shape with the next — a shape that cannot land in both of them. Then make the two ends share one more shape on top of that. Whichever end is forced to lock has to spend that shape inside the chain, so any cell outside the chain that sees every place it could still go in either end cannot hold it.
What an ALS chain looks like
A pair of almost locked sets joined by one restricted shape is as far as the primer goes, and it is already enough to eliminate. A chain is the same argument passed along. Set one hands set two a shape it cannot keep; set two locks and hands set three a different shape it cannot keep; and the relay ends at whichever end of the chain was not the one to start it. Nothing new is being claimed at any step — only the same "N cells, N+1 shapes, take one away and it locks" you already know, applied three times in a row.
Read it as a relay. The two cells of the tinted pair hold the plus, the ring and the diamond between them — two cells, three shapes, so nothing is settled. Take the plus off them and they are two cells over two shapes and locked. That is exactly what the single cell in the top-left box does if it turns out to be the plus, because it sees the only place the pair could put one. And a locked pair has to use its diamond, which is in the right-hand cell of the pair — in the same row as the third group, which would then lose its own diamond and be left as the ring. Follow it the other way and the cell in the top-left box is the ring instead. Either way one of the two end cells is a ring, and the cell that sees both of them is not.
How to use an ALS chain
- Start from a pair that goes nowhere. Take two almost locked sets sharing a restricted shape — the pairing from the primer. If the two of them already share a second shape and something outside sees all of it, you are done and this page is not needed. Chains are what you reach for when they share only the one.
- Hang a third set off the far end. Look for another almost locked set that shares a restricted shape with the second one, and make sure it is a different shape from the first link. A set cannot spend the shape its neighbour just took off it, and a chain that repeats a link goes dead there. The sets must also share no cells at all.
- Check the two ends for a shared shape. Now compare the first set with the last. They need one shape in common that is neither of the two links. That shape is the payoff — call it z. If the ends have no such shape, the chain is real but eliminates nothing, so move the third set and try again.
- Strike z outside the chain. Mark every cell in the first set and the last set that can still hold z. Any cell outside the whole chain that sees all of them loses z. One end or the other is forced to keep it, and a cell seeing both ends cannot survive either case.
Three sets is where this stops being a wing and starts being a chain, and it is also where form9 stops looking: a fourth set is a fourth group of cells to tint and a fourth count to hold in your head, and by then a forcing chain on one of the same cells is usually easier to see and easier to check.
ALS chains vs the wings
An XY-Wing is three bivalue cells with a pivot every wing can see, and a WXYZ-Wing is the same idea with a four-candidate hub. Both need that hub. A chain does not: the middle set only has to see the set before it and the set after it, so the two ends can sit in different boxes on opposite sides of the grid and still argue. That is the whole reason to learn it — the wings run out of positions long before the sets do.
Find ALS Chains in your own grid
Paste a puzzle and this page will look for ALS Chains in it — the same pattern shown above, in your grid instead of ours. Nothing is uploaded; it runs in your browser.
Frequently asked questions
How is this different from ALS-XZ?
ALS-XZ is the two-set case: one restricted shape between the sets, one more shape shared by both, done. An ALS chain is the same rule with sets in between, each one locking and passing a different shape along. Two sets is the pairing the primer teaches; the board above needs all three, and no pairing among them reaches the same mark on its own.
Why must consecutive links use different shapes?
Because a set in the middle of the chain has just lost the incoming shape — that is what made it lock. Asking it to then spend that same shape is asking it to hold a shape it no longer has, and the relay stops right there. Every step forward has to be a shape the set still owns, which is any of its remaining values except the one it just gave up.
Can the sets overlap?
No, and it is not a technicality. Two sets sharing a cell would each be counting that cell's shape as their own, and the "restricted" test — every place one set could put the shape sees every place the other could — is meaningless when a cell has to see itself. form9 only reports chains whose sets are completely disjoint.
How long can a chain get?
In theory as long as you like. In practice form9 stops at three sets, for the same reason the naked quad page stops at four cells: a fourth set is a fourth group to tell apart on one grid, and a figure nobody can read teaches nobody anything. Longer chains exist in solver literature; they are not something you find by eye.
When is it worth hunting for one?
Late, and only when a straight pairing has already failed — most grids that hold a chain also let something shorter reach the same marks. The board above is deliberately not one of those: none of the twenty-four easier techniques form9 knows can strike that ring, and neither can any two of its three sets on their own. 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.