An X-chain follows one digit along links that alternate strong and weak, starting and ending on a strong link. A strong link joins the only two places for the digit in a house. A weak link joins two cells that see each other. At least one end of the chain holds the digit, so a cell that sees both ends cannot.
How to spot it
An X-chain works on one digit, like simple coloring. It uses two kinds of link.
A strong link joins the only two places for the digit in a house. If one of them is not the digit, the other one is.
A weak link joins any two cells that hold the digit and see each other. They cannot both be the digit, so if one is, the other is not. If one is not, the weak link tells you nothing.
Build a path that takes the two kinds in turn, strong, weak, strong and so on, and make the last link strong as well. Then look for a cell holding the digit that sees both ends of the path. That cell loses the digit.
Start on a strong link
Take the 9. Column five has only two places for it, row one and row six. That is a strong link. One of the two cells is the 9.
Step across a weak link
Row six has three places for the 9, in columns two, five and six. So row six, column five and row six, column two are not a strong pair. They do see each other, though, and only one of them can be the 9. That makes a weak link, drawn dashed.
Alternate to a strong link at the end
Box four has two places for the 9, so a strong link takes the chain to row four, column three. Column three has three places, so the next link is weak and goes down to row eight, column three. Row eight has two places, and a last strong link ends the chain at row eight, column nine. Six cells and five links in five different houses.
Read the chain from one end
Suppose row one, column five is not the 9. Then column five puts the 9 in row six, column five. The weak link in row six rules out row six, column two. Box four puts the 9 in row four, column three. Column three rules out row eight, column three. Row eight puts the 9 in row eight, column nine. So if one end is not the 9, the other end is. At least one of them holds it.
Strike the 9 that sees both ends
Row one, column nine shares row one with one end and column nine with the other. Whichever end holds the 9, it is in a house with row one, column nine, so the 9 comes out of that cell. Box three now has one place left for the 9, row two, column eight. That is a hidden single. Before the strike the grid had no single of either kind.
Why both ends must be strong
Across a strong link, a cell that is not the digit makes its partner the digit. Across a weak link, a cell that is the digit makes its partner not the digit. Put them in turn and each one hands its result to the next. "Not the 9" turns into "the 9" at the next cell, then back into "not the 9", all the way along.
So the chain has to start and end on a strong link. Stop at row eight, column three, right after the weak link in column three, and all you know is that if row one, column five is not the 9, row eight, column three is not either. That does not put the 9 anywhere. The strong link in row eight is needed to turn it into a 9 at the far end.
Read the same chain from the other end and you get the mirror result. So one of the two ends always holds the 9, and here both do. In the solution the 9 is in row one, column five and in row eight, column nine.
A weak link where a strong one has to be
Start at row four, column three and run the chain backwards to row one, column five. Then go on along row one. Row one, column nine sees row one, column five, so a weak link to it is fine. From there the chain needs a strong link, and column nine looks like one, down to row seven, column nine.
It is not. Column nine has a third place for the 9, in row eight. Treat it as strong anyway and the ends are row four, column three and row seven, column nine. Row seven, column three sees both, so you would strike its 9. In the solution that is exactly where the 9 of row seven goes.
Any two cells that see each other make a weak link. A strong link needs a house with exactly two places for the digit. Count them before you use one.
Two places. If one is not the 9, the other is, so the link can sit anywhere in the chain.
Three places. Rows one and seven cannot both be the 9, but neither has to be, because row eight is left.
What counts as a weak link
A weak link only asks that the two cells see each other. The house can have three places for the digit or six. Row six and column three in the chain above have three each.
A strong link can stand in a weak spot too. Two places in a house still cannot both hold the digit. So after a strong link, the next cell can be any cell that holds the digit and sees the current one, whether or not the two also form a strong link.
This is also where the X-chain parts from simple coloring. Coloring only uses strong links, so two colors are enough and one of them is true everywhere. In this grid the 9 is in neither cell of the weak link in column three, between rows four and eight. Across a weak link, all you learn is that the two cells are not both the 9.
Pick a digit. Join cells with links that alternate strong and weak, and start and end on a strong one. At least one end then holds the digit, so strike it from any cell that sees both ends. A strong link needs exactly two places in its house. A weak link only needs two cells that see each other.
Where this sits on the path
An X-chain places nothing. It removes candidates and leaves the placing to the singles. On the grid above it made one hidden single.
It is the second step of the chapter on chains and coloring. Simple coloring followed one digit through strong links. The X-chain follows one digit too, but a weak link lets it go on where coloring stops. On this grid nothing earlier on the path removes a candidate, simple coloring included.
X-chains with cell groups come next. They let one end of a link be two or three cells that share a box and a row or column. Later in the expert tier, an XY-chain alternates its links the same way, but it uses the two digits inside each cell instead of one digit across cells.
This chain runs from row four, column six to row two, column eight. Which 9 does it remove?
Find the two ends. Then look for a cell holding a 9 that shares a house with each of them.
Recap
- Look for
- One digit with exactly two places in some houses, and cells that see each other between them, so the links can alternate strong and weak.
- Scope
- One digit. The cells of the chain, and any other cell holding that digit that sees both ends.
- Result
- The digit comes out of every cell that sees both ends of the chain.
- Tell-tale
- Strong links on one digit that almost join up, with a gap two cells in one house can bridge.
- If it fails
- If no cell sees both ends, extend the chain or start from another strong link. A link in a strong position needs exactly two places in its house.