Chains, Colouring & Uniqueness
XY-Chain
An XY-Chain is a run of bivalue cells — cells with exactly two candidates — in which each cell sees the next and shares with it the digit that links them, and whose two ends both hold the same digit Z. If the first cell is not Z it must be its other digit, which forces every following cell in turn until the last cell has to be Z; so Z is in the first cell or the last, and it can be removed from every cell that sees both ends. An XY-Wing is the shortest XY-Chain, three cells long.
What you need to see on the grid
Look for an XY-Chain once singles, subsets, fish and wings have stopped producing moves and the grid still holds a good number of bivalue cells: the chain lives entirely among them. Four terms carry the whole method:
- Bivalue cell
- A cell with exactly two candidates left. Every cell of the chain must be one: rule out one digit and the other is forced. A cell with three candidates breaks the chain, because ruling one out still leaves two.
- A link
- Two consecutive cells of the chain see each other — same row, column or box — and both hold the digit being passed along. When one cell is forced to that digit, its neighbour cannot be it, so the neighbour takes its own other digit and passes that on.
- The two ends and Z
- The first and the last cell of the chain both hold the same digit, Z. You start by assuming the first cell is not Z; the chain is complete when that assumption forces the last cell to be Z. Z is the only digit the chain says anything about.
- Cells that see both ends
- The only cells an XY-Chain clears: they share a unit with the first cell and a unit with the last cell, and still list Z. A cell that sees only one end, and every cell of the chain itself, keeps its candidates.
How to apply it, step by step
One example runs through all five steps: a chain on the digit 7, part-way through a hard puzzle. Every diagram shows the whole grid, but only the cells a step needs are filled in; the others are left blank and hatched. The cells shown have already lost their other candidates to earlier moves elsewhere in the grid, which is why so many of them are down to two.
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Step 1List the bivalue cells and pick a start
An XY-Chain is built only from bivalue cells, so begin by spotting them. In this diagram six cells hold exactly two candidates: R2C2, R2C7, R3C9, R8C2, R8C4 and R8C9. R2C4 and R8C6 have three, so they can never be links — but keep an eye on them, they may be what the chain clears.
Pick one bivalue cell as the start and choose one of its two digits as Z, the digit you hope to meet again at the far end. Here the start is R2C2 {2,7}, with Z = 7. Now assume the start is not 7: then it must be its other digit, 2. That one assumption is all the chain needs.
Eight cells matter in this example. The six outlined ones are bivalue; R2C2, numbered 1, is the start. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - Bivalue cell (two candidates)
- Chain cell, numbered in order
- Not needed for this step
Filled in: cells R2C2, R2C4, R2C7, R3C9, R8C2, R8C4, R8C6 and R8C9. The other 73 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 2Walk the chain one link at a time
From the start, look for a bivalue cell that sees it and holds the digit the start was just forced to. R2C2 is 2, and R2C7 {2,5} sits in the same row: it cannot be 2 as well, so it is 5. R3C9 {5,9} shares box 3 with R2C7, so it cannot be 5 and is 9. R8C9 {3,9} is in column 9 with R3C9, so it is 3.
At every link check two things: the two cells see each other, and the next cell holds the digit just forced. That digit is ruled out in the next cell, so the cell takes its other digit — and that is the digit you carry to the following link.
The first four cells of the chain, numbered in order: a row link, a box link and a column link. If R2C2 is not 7, they become 2, 5, 9 and 3 in turn. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - The start, assumed not to be 7
- Chain cell, numbered in order
- Candidate 7, the digit studied
- Not needed for this step
Filled in: cells R2C2, R2C7, R3C9 and R8C9. The other 77 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 3Stop at a cell that is forced to Z
Keep walking until the digit you are forced onto is Z. R8C4 {3,7} shares row 8 with R8C9, which was just forced to 3, so R8C4 is 7 — the digit we started from. The chain is complete: R2C2, R2C7, R3C9, R8C9, R8C4, five cells and four links. Read as one sentence, it says: if R2C2 is not 7, then R8C4 is 7.
If a walk lands on a cell whose forced digit is not Z, the chain is not finished: go on to another link, or step back and try a different one. The app's hint engine looks for chains of up to 13 cells; on paper, four to six is a realistic target. With exactly three cells you would have an XY-Wing — the pincers are the ends and the pivot is the cell in between.
Technique: XY-Chain
The whole chain, numbered 1 to 5. Both ends, R2C2 and R8C4, hold 7. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - An end of the chain, holding 7
- Chain cell, numbered in order
- Candidate 7, the digit studied
- Not needed for this step
Filled in: cells R2C2, R2C7, R3C9, R8C4 and R8C9. The other 76 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.
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Step 4Either end is Z: clear Z from cells that see both
Now the two cases. Either R2C2 is 7, or it is not — and then the chain makes R8C4 7. There is no third case, so a 7 sits in R2C2 or in R8C4, even though we do not know which. A cell that sees both ends sees that 7 either way, so it cannot be 7.
R2C4 shares row 2 with R2C2 and column 4 with R8C4; R8C2 shares column 2 with R2C2 and row 8 with R8C4. Both lose 7. R8C6 keeps it: it sees R8C4 but not R2C2, and when the 7 lands on R2C2 nothing stops R8C6 from being 7. The chain cells keep all their candidates, and nothing is placed yet.
Technique: XY-Chain
The chain and the three other cells that list 7. R2C4 and R8C2 see both ends and lose 7; R8C6 sees only R8C4 and keeps it. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 R9 - An end of the chain, holding 7
- Chain cell, numbered in order
- Candidate 7, the digit studied
- Candidate removed
- Not needed for this step
Filled in: cells R2C2, R2C4, R2C7, R3C9, R8C2, R8C4, R8C6 and R8C9. The other 73 cells are left blank: they are not needed for this step. Removed: 7 from R2C4 and R8C2.
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Step 5Collect what the elimination unlocks
An XY-Chain never places a digit by itself, so look at the cells it just cleared. R8C2 was {6,7}; without 7 it holds only 6 — a Naked Single. R2C4 is down to {1,8}: a new bivalue cell, and new material for the next chain.
Check every cell that lost a candidate before you move on — that is where an XY-Chain pays you back.
Technique: Naked Single
The two cleared cells after the XY-Chain. R8C2 can only be 6; R2C4 is now bivalue. C1 C2 C3 C4 C5 C6 C7 C8 C9 R1 R2 R3 R4 R5 R6 R7 R8 6 R9 - One candidate left
- Candidate 6, the digit studied
- Digit placed
- Not needed for this step
Filled in: cells R2C4 and R8C2. The other 79 cells are left blank: they are not needed for this step. Placed: R8C2 = 6.
Common mistakes
Linking two cells that do not see each other
Sharing a digit is only half of a link; the two cells must also share a row, column or box. R1C1 {2,7} and R5C6 {2,8} both hold 2, but one being 2 says nothing about the other, and the chain breaks there. Check the unit for every pair of consecutive cells, not just the digits.
Passing through a cell with three candidates
A link only forces something when the cell has two candidates: rule one out and the other is left. In a cell such as {1,7,8}, ruling out 7 still leaves 1 or 8, so nothing is passed on and every conclusion after it is unproved. Count the candidates of every cell you add — exactly two.
Eliminating from a cell that sees only one end
The chain guarantees Z at one end or the other, never at a named one. R8C6 in the example sees R8C4 but not R2C2, so when the 7 is at R2C2 it may still be 7. Remove Z only where a cell sees both ends.
Recent daily puzzles that needed this technique
Keep going
Learn the move, not just the digit
In a real game, the hint in the app shows the next cell you can fill and explains why. When an XY-Chain was needed to get there, the explanation names XY-Chain, or another technique of the same level it used on the way, as the reason a candidate is ruled out. It does not draw the chain for you: the steps above are how you find it.