A cell with three candidates, the pivot, sees two cells with two candidates, the wings. Each wing holds two of the pivot's digits, and one digit is in all three cells. Whichever digit the pivot takes, that shared digit lands in one of the three, so any cell that sees all three loses it.
How to spot it
The Y-wing needed a pivot with two candidates. Give that pivot a third candidate, the digit its wings share, and you have an XYZ-wing.
Pick a cell with three candidates, say X, Y and Z. That is the pivot. Among the cells it sees, look for one with exactly X and Z, and another with exactly Y and Z. Those are the wings. Every digit in the wings comes from the pivot, and Z is the only digit all three cells hold.
The strike is narrower than in a Y-wing, because a target has to see the pivot as well as both wings. That only works when one wing shares the pivot's box and the other shares its row or column. The targets are where that row or column crosses the box, two cells at most besides the pivot. If all three cells sit in one house, they are a naked triple, which comes earlier on the path.
Pick a pivot with three candidates
Only four empty cells on this board have three candidates. Take the one in row eight, column four. It holds 1, 2 and 9.
Find two wings that share one digit
The pivot sees the cell in row four of its own column, which has 1 and 2. It also sees the cell in column six of its own row and box, which has 2 and 9. Each wing holds two of the pivot's digits, and the only digit in all three cells is the 2. That makes an XYZ-wing on 2. The cell in row seven, column four also sees the pivot and has two candidates, 2 and 8. The pivot has no 8, so it cannot be a wing.
Follow all three of the pivot's digits
If the pivot is 1, the wing in row four shares its column and cannot be 1, so it is 2. If the pivot is 9, the wing in column six shares its row and cannot be 9, so it is 2. If the pivot is 2, the 2 is in the pivot. There is no fourth case. One of the three cells holds the 2, and you do not need to know which.
Strike the 2 from cells that see all three
A target has to share a house with each of the three cells, the pivot included, because the 2 might be in the pivot. The pivot and the wing in column six share box eight. The pivot and the wing in row four share column four. Box eight meets column four in three cells. One is the pivot, one is a given 3, and the third is row seven, column four. It loses its 2.
Read what the strike left behind
The cell in row seven, column four had 2 and 8. Now it is an 8, a naked single. Before the strike this grid had no single of either kind. Nothing earlier on the path removed a mark either, from pointing pairs through to the Y-wing. The XYZ-wing is the first technique that does anything here.
Why the target must see the pivot
The argument splits on the pivot, as it did for the Y-wing, but now there are three cases, the ones the third step walked through. Each case puts the 2 in a different cell of the pattern, so only a cell that sees all three sees it every time.
In a Y-wing the pivot never holds the shared digit, so a target only has to see the two wings. Here the third case puts the 2 in the pivot itself, and a cell that cannot see the pivot is not covered by that case.
On this grid the third case is the true one. The pivot is 2, the wing in row four is 1 and the wing in column six is 9. Neither wing ends up with the 2. The strike still holds, because row seven, column four sees the pivot.
A cell that sees only the wings
Go back to the board before the strike. The cell in row six, column six has 2 and 3. It shares box five with the wing in row four and column six with the other wing. It sees both wings, which is all a Y-wing would ask for.
It does not see the pivot. Row six, column six is in a different row, column and box from row eight, column four, so the third case, a pivot of 2, says nothing about it. Strike its 2 anyway and you remove the right answer. In the solution the pivot is 2, and row six, column six is 2 as well.
Before you strike, check the pivot. A target has to see all three cells.
It shares column four with the pivot and the wing in row four, and box eight with the pivot and the wing in column six. Its 2 goes.
It shares a box with one wing and a column with the other, but nothing with the pivot. A pivot of 2 leaves it alone, and on this grid it is 2.
Three more that strike nothing
This board holds three more XYZ-wings, and none of them removes anything.
In row nine, the cell in column five has 1, 6 and 9. The cells in columns seven and nine have 6 and 9, and 1 and 6. That is an XYZ-wing on 6 with all three cells in one row. A cell that sees all three would have to be in row nine, and row nine has no other empty cell. Three cells sharing three digits in one house are a naked triple, and this one has nothing left to clear. Column nine holds the same shape in rows two, eight and nine.
The third has its pivot in row seven, column seven. It is the check at the end of this page.
Look for a cell with three candidates. Find two cells it sees with two candidates each, made only of the pivot's digits and sharing one of them. Strike that digit from every cell that sees the pivot and both wings. A cell that sees only the wings keeps it.
Where this sits on the path
An XYZ-wing places nothing. It removes candidates and leaves the placing to the singles. On the grid above it made one naked single, in row seven.
It is the second of the wings. The Y-wing splits on a pivot with two candidates. The XYZ-wing gives the pivot a third, and pays for it with a smaller strike zone.
The W-wing comes next. It drops the pivot and joins two cells through a digit with two places in a house.
On the board before the strike, the cell in row seven, column seven has 2, 3 and 6. It sees the cell in row two, column seven, with 2 and 6, and the cell in row seven, column one, with 3 and 6. Both wings carry a 6. Which 6 can you strike?
A target has to see the pivot and both wings. Look for a cell inside the houses of all three.
الخلاصة
- ابحث عن
- A pivot with three candidates that sees two wings with two candidates each. Every wing digit comes from the pivot, and the two wings share one digit.
- النطاق
- Every cell that sees the pivot and both wings. The pivot and the wings keep their candidates.
- النتيجة
- The shared digit comes out of every cell that sees all three.
- العلامة
- A Y-wing that does not quite fit, because the pivot has a third candidate, the one the wings share.
- إذا لم تنجح
- If no cell sees all three, there is nothing to strike, even when some cells see both wings. Try another pivot, or go back to the cells with two candidates for a Y-wing.