XY-Chain — one digit, walked along a chain of pairs

The short version

An XY-Chain is a line of cells that each hold exactly two candidates. Walk from one to the next: each cell must share one of its digits with its neighbour, and must be visible to it.

Now look at the two ends. If they share a digit — say both carry an 8 — then that digit is decided.

Why: the first cell has only two candidates. If it takes the 8, an end is the 8. If it takes the other one, the second cell cannot repeat that digit, so it is forced to its other candidate, and that forces the next, all the way down until the far end is left with the 8.

Either way, one of the two ends holds the 8. So any cell that can see both ends cannot be 8.

Nothing was assumed. Every step is "this cell has two candidates and one of them is taken, so it must be the other".

Why the ends are enough

The chain looks like a long argument, but only its two ends matter.

end A = {x, d}  —  {d, e}  —  {e, f}  —  …  —  {g, x} = end B

Take the left end. If it is x, done. If it is not x, it is d, and d is now used — so the next cell, which is {d, e}, has to be e. That forces the cell after it, and so on, until the last cell has no choice but to be x.

So x is in A or in B. That is the whole theorem, and it never needs more than two candidates per cell — which is exactly why the chain is restricted to those cells.

A real board

The chain here is four cells long, and it runs down column 4 before turning into row 5:

• R1C4 holds {4, 8}.
• R7C4 holds {1, 4} — it sees R1C4, and they share the 4.
• R5C4 holds {1, 9} — it shares the 1 with R7C4.
• R5C2 holds {8, 9} — it shares the 9 with R5C4.

Both ends carry an 8: R1C4 and R5C2. Now test the first cell.

• If R1C4 = 8, an end is the 8, and we are done.
• If R1C4 = 4, then R7C4 cannot be 4, so R7C4 = 1. Then R5C4 cannot be 1, so R5C4 = 9. Then R5C2 cannot be 9, so R5C2 = 8.

Either way, one of R1C4 and R5C2 is 8. R1C2 sits in row 1 with R1C4 and in column 2 with R5C2 — it sees both ends, so it cannot be 8.

Step through the board above to see it happen: the chain lights up one cell at a time, and only then does the 8 come off R1C2.

How to find one

You do not search for a chain. You extend one.

1. Start at a cell with two candidates. Any of them. Beginners pick the one with the most neighbours.
2. Look for a cell it sees that shares a digit. The neighbour must also hold exactly two candidates, and one of them must be a digit you already have.
3. Step across. The shared digit is now "used", so the neighbour's other digit becomes the digit you carry forward.
4. Repeat until the carried digit matches something at the start. If the number you are carrying appears in the starting cell's pair, the chain closes and the elimination is live.
5. Check what sees both ends. The digit that both ends carry comes off every cell that is visible from both.

The fourth step is the one that decides everything. A chain that stops without closing is just a list of cells.

XY-Chain, XY-Wing, forcing chain

These three get confused constantly, and the differences matter when you are deciding what to look at.

The XY-Wing is a chain of three cells with an extra condition — the middle cell has to hold all three digits. The forcing chain is more powerful and much slower: it leaves the two-candidate world entirely and follows every consequence of an assumption. In practice, look for chains first, and fall back to forcing when the board will not move.

The traps

Using a cell with three candidates. The logic only works while every cell has exactly two. One three-candidate cell breaks the "must be the other one" step, and the rest of the chain is worthless.

Reading the share wrong. Two cells can see each other and still share nothing. Check the actual candidates, not the feeling that they are related.

Forgetting that the ends must share a digit. A chain that ends on a different digit proves nothing. This is the most common way an XY-Chain appears to work when it does not.

Building the chain out of eliminated candidates. If you have already removed a candidate somewhere on the chain, the chain has to be built from the reduced list — and the diagram in your head is now one step out of date. Start from the candidates as they actually stand.

Going too long. Every extra cell is another chance to misread a pair. A four-cell chain you can verify is worth more than an eight-cell chain you cannot.

Where to use it

XY-Chains are the first technique on this site that is genuinely hard to spot by eye, and they are also the first that reliably cracks boards where every pattern technique has failed. The order to work in is: patterns, then fish, then wings, then chains, then forcing.

If you find yourself looking for a chain on every board, you are skipping the cheaper moves. Most Expert boards fall to a skyscraper or a W-Wing first — and those take seconds rather than minutes.

How often does this technique actually show up? See its measured numbers →