An XY-chain runs through cells that each have exactly two candidates. Inside each cell, if one candidate is false the other is true. Two cells that see each other can't both hold a digit they share. When both ends of the chain carry the same digit, one end holds it, so a cell that sees both ends loses it.
How to spot it
Start with the cells that have exactly two candidates. Late in a puzzle there are plenty, and this grid has sixteen.
Pick one, and pick the digit you hope to remove, one of its two. Assume the cell is not that digit. Then it is the other one, and that other digit is ruled out of every cell it sees.
Look for a cell it sees that has two candidates, one of them the digit you just ruled out. That cell must take its other candidate. Keep going, one cell at a time. When a cell's other candidate is the digit you started with, stop. The first cell and the last cell are the ends, and any cell that sees both of them loses that digit.
Start in a cell with two candidates
Row four, column one has only a 1 and a 3. Take the 1 as the digit to remove. If this cell is not the 1, it is the 3. A solid line inside the cell stands for that.
Pass the other digit along
Row four, column three shares row four with it and has a 2 and a 3. If the first cell is the 3, this one can't be, so it is the 2. Column three then leads to row eight, column three, which has a 2 and a 9. It can't be the 2, so it is the 9. A dashed line marks each step, on the digit that is passed along.
Stop when the 1 comes back
Row eight, column nine shares row eight with the last cell and has a 9 and a 1. It can't be the 9, so it is the 1. The chain started on a 1 and ends on a 1. Put together, if row four, column one is not the 1, row eight, column nine is.
Strike the 1 that sees both ends
Row eight, column one shares column one with the first end and row eight with the second. One of the two ends holds the 1, and either way this cell sits in a house with it. So the 1 comes out of row eight, column one.
Collect the singles
Row eight, column one is left with a 6, a naked single. That leaves column one with one place for the 1, row four, column one, and row eight with one, column nine. Box seven's last place for the 1 is row nine, column two. Those are three hidden singles. Before the strike there was no single on the board.
Why one end holds the 1
Run the chain from the other end. If row eight, column nine is not the 1, it is the 9. Row eight then makes row eight, column three a 2, column three makes row four, column three a 3, and row four makes row four, column one the 1. So whichever end you start from, if that end is not the 1, the far end is.
In the solution both ends hold the 1, though the chain proves only that one of them does.
A Y-wing is a three-cell XY-chain
A Y-wing is the same idea with three cells. The pivot is the middle cell, the wings are the ends, and both wings carry the digit that goes.
Stop this chain after three cells and you get a pivot with two wings. Row four, column three is the pivot, and its wings are row four, column one and row eight, column three. They carry a 1 and a 9, not the same digit, so the three cells strike nothing. A fourth cell, row eight, column nine, turns the 9 back into a 1. The in-game hint calls a chain of four or more cells an XY-chain, since three cells are already a Y-wing.
How it differs from an X-chain
An X-chain stays on one digit. Its strong links run between two cells, where a house has only two places for that digit.
In an XY-chain the strong links sit inside cells, where a cell has only two candidates. Every link between cells is weak, so two cells only need to see each other and share a digit. That digit changes as you go. Here it is 3 along row four, 2 down column three and 9 along row eight. Only the ends have to carry the same digit.
Both chains end the same way. One end holds the digit, so any cell that sees both ends loses it. Later in this chapter, longer AICs use both kinds of strong link in one chain.
Row eight, column three has a 2 and a 9. Rule out the 2 and the 9 is left.
Row seven, column three has a 2, a 3 and a 9. Rule out the 9 and two digits are left, so nothing passes on.
A cell with three candidates
Try a chain on the 1 from row four, column two, which has a 1 and a 2. If it is not the 1, it is the 2. Column two then makes row seven, column two a 7. Row seven takes the 7 away from row seven, column nine.
That cell has a 4, a 7 and a 9. Without the 7 it can still be a 4 or a 9, and the chain has no right to pick. Pick the 4 anyway, and box nine makes row nine, column eight a 1. Both ends now carry a 1, and row nine, column two sees both. You would strike its 1.
In the solution row nine, column two is the 1. It is one of the hidden singles the real chain opened. Before you add a cell to a chain, check that it has exactly two candidates.
Start in a cell with two candidates and assume it is not the digit you want to remove. Pass its other digit to a cell it sees that also has two candidates, and keep going. When a cell's other digit is the one you started with, strike that digit from every cell that sees both ends. Every cell in the chain needs exactly two candidates.
Where this sits on the path
An XY-chain never places a digit by itself. The singles it opens do that, and here one strike opened four.
It belongs to the chapter on chains and coloring, after short AICs. On this grid nothing earlier on the path removes a candidate, Y-wings, X-chains and short AICs included.
XY-chains with cell groups come next.
This chain runs from row eight, column three to row nine, column seven, and both ends carry a 2. Which 2 does it remove?
Find a 2 that shares a house with each end.
الخلاصة
- ابحث عن
- A path of cells with exactly two candidates, each seeing the next and sharing a digit with it, whose two ends carry the same digit.
- النطاق
- The cells of the chain, and any cell holding the end digit that sees both ends.
- النتيجة
- The end digit comes out of every cell that sees both ends.
- العلامة
- Many cells down to two candidates, often near the end of a puzzle, with pairs of them sharing digits across rows and columns.
- إذا لم تنجح
- Check every cell in the chain for a third candidate. If no cell sees both ends, start from another cell or another digit.