The pivot kept a third digit

An XYZ-wing is three cells. The pivot has three candidates. Each wing has two, and each wing uses a different pair drawn from the pivot. Both wings see the pivot. They do not have to see each other.

An XY-wing is that same shape with only two candidates on the pivot. Here the pivot is row 5, column 1, and it can be 1, 3, or 5. One wing is row 4, column 2, in the same box, and it can be 3 or 5. The other wing is row 8, column 1, down the column, and it can be 1 or 3. The wings share 3. The pivot can be 3 as well. That extra seat is the whole technique.

Every story puts a 3 in the pattern

If the pivot is 1, row 8, column 1 cannot be 1, so that wing is 3. If the pivot is 5, row 4, column 2 cannot be 5, so that wing is 3. If the pivot is 3, the digit is already in row 5, column 1. The 3 lives in the pivot or in a wing. A cell that sees all three cannot be 3.

On an XY-wing the pivot cannot hold the digit the wings share, so the victim only has to see the wings. Here the pivot is allowed to be 3. A cell that misses the pivot can still be 3, on the story where the pivot takes it. I learned that by deleting the 3 and then watching the pivot want it.

Three cells, and the middle one is busier

Row 5, column 1 is the pivot. It sees both wings, and the wings do not see each other. Step through. The 3 leaves the one cell that can see all three.

See the pattern

Look at these three cells.

Who sees whom on this board

Row 4, column 2 sits in box 4 with the pivot. Row 5, column 2 is 2 and row 5, column 3 is 9, so the link is the box. Row 8, column 1 shares column 1 with row 5, column 1. Row 6, column 1 is already 8 and row 7, column 1 is 9, which is why the second wing is down in row 8 rather than next door.

Row 4, column 2 and row 8, column 1 are different rows, different columns, and different boxes. Box 4 and box 7. They do not see each other. The pivot is the cell that has to see both, and it does.

Row 4, column 1 can be 1, 3, 4, or 5. It sees row 5, column 1 down column 1 and inside box 4. It sees row 4, column 2 along the row. It sees row 8, column 1 down the same column. All three. The 3 leaves. 1, 4, and 5 stay. Nothing is placed.

The cell that only sees the wings

Row 8, column 2 can be 3 or 8. It sees row 8, column 1 along the row, and it sees row 4, column 2 down column 2. Both wings. It does not see row 5, column 1. The pivot can still be 3, so row 8, column 2 keeps the 3. That is the XY-wing deletion on the wrong pattern. I have crossed it off, and then put it back.

A naked triple needs the three pattern cells in one house. These three are split. Row 8, column 1 is in box 7, and row 5, column 1 and row 4, column 2 are in box 4. The candidates together are 1, 3, and 5, which looks tidy on a list. The houses are what make it a wing.

One step past the shorter wing

XYZ-wings are Expert. The next page is the W-wing, and the swordfish comes after that. They stay short of coloring and XY-chains. XYZ-wings do not appear in the 1,483 NYT puzzles. Hard stops at triples.

I look for one when a two-digit cell is touching a three-digit cell, and a second two-digit cell takes the pair that is left. Two candidates on the pivot means the shorter wing, and that page is the one to use. Three candidates on the pivot means the victim has to see all three cells, pivot included.

Practice this technique

That jumps a puzzle forward until XYZ-wing is the next move. Or start a full Expert puzzle.

Next harder: W-wing