Techniques · Expert
Two-string kite
A row and a column, tied in one box
A two-string kite is one digit with two strong links that are not parallel. A row has the digit in exactly two cells. A column has it in exactly two other cells. One cell from the row and one from the column sit in the same box, and that box is the knot. The other two cells are the ends. One of those ends is the digit. A cell that sees both ends loses it.
On this board the digit is 8. Row 5 holds it in row 5, column 5 and row 5, column 7. Column 8 holds it in row 2, column 8 and row 6, column 8. Row 5, column 7 and row 6, column 8 are both in box 6. The ends are row 5, column 5 and row 2, column 8. The name is a lot of string for two lines and a box. The picture is simpler than the name.
The houses, before the story
Row 5 is mostly filled. Row 5, column 1 is 3, row 5, column 3 is 4, row 5, column 4 is 7, row 5, column 8 is 6, and row 5, column 9 is 5. Row 5, column 2 can be 1 or 9. Row 5, column 6 can be 1 or 2. Neither lists an 8. The two homes are row 5, column 5, which can be 2, 8, or 9, and row 5, column 7, which can be 2 or 8.
Column 8 is the other string. Row 1, column 8 is 3. Row 4, column 8 is 4, row 5, column 8 is 6, and row 8, column 8 is 1. Row 3, column 8 can be 2 or 7. Row 7, column 8 and row 9, column 8 can each be 5 or 9. The only 8s left in the column are row 2, column 8 (2, 7, or 8) and row 6, column 8 (2 or 8).
Why one end has to be 8
A box holds 8 once, so the two links cannot both put their 8 in box 6. If row 5, column 7 is 8, then row 6, column 8 is not, and column 8's 8 is row 2, column 8. If row 6, column 8 is 8, then row 5, column 7 is not, and row 5's 8 is row 5, column 5. One of the ends is 8. You never learn which.
Row 2, column 5 can be 2, 4, 6, 7, or 8. It shares row 2 with row 2, column 8, and it shares column 5 with row 5, column 5. Both ends. The 8 leaves. 2, 4, 6, and 7 stay. The step places nothing. A kite that only cleans one note is still a kite. I used to wait for a placement and miss the move.
Row 5 meets column 8
The two 8s in the row and the two 8s in the column meet in box 6. Step through. One 8 leaves the cell that sees both ends.
See the pattern
Look at the 8s in this row and this column.
Parallel lines are a different fish
A skyscraper uses two rows, or two columns. This uses one of each. If I catch myself checking whether the free ends share a column, I am still on the fish. The tie here is box 6, and the lines cross.
The deletion sits on the ends, not on the two cells that share the box. Row 5, column 7 and row 6, column 8 are the knot. They keep their 8s. Rubbing the 8 off either of them throws the link away.
A cell has to see both ends. Row 2, column 7 can be 1, 2, or 8. It sees row 2, column 8 along the row, and it does not see row 5, column 5. The 8 stays. One end is the near miss I still make, on a cell that shares a row with one end and nothing with the other. If it misses the second end, it is not this move.
Expert, between the skyscraper and the wing
Two-string kites are Expert. They sit after the skyscraper and before the XY-wing. Mental is still reserved for coloring and XY-chains. No puzzle in the NYT archive needs a two-string kite. That set never goes beyond triples.
I reach for it when one digit has a two-cell row and a two-cell column, and one cell from each falls in the same box. An X-wing is the simpler fish when the two lines are parallel and the ends line up. I take that first, then the skyscraper, and the kite is what is left when the links refuse to be parallel.
That jumps a puzzle forward until Two-string kite is the next move. Or start a full Expert puzzle.