Techniques · Expert
Skyscraper
The fourth corner slid over
A skyscraper is two strong links on one digit that share one end and miss the other. On this board the digit is 5. Row 3 holds a 5 in exactly two cells, and row 8 holds a 5 in exactly two cells. Both rows use column 3. The other two cells, the roof, sit in column 7 and column 9. They do not line up.
An X-wing would have wanted those free ends in the same column. They refused. The rectangle is still there if you let one corner wander, and the digit still has to live on that roof.
Why one roof cell is 5
Column 3 can take the 5 only once. Row 3, column 3 and row 8, column 3 cannot both be 5, so at least one roof cell is the digit. If row 3, column 3 is not 5, then row 3, column 7 is 5. If row 8, column 3 is not 5, then row 8, column 9 is 5. The two base cells cannot fail together. One of the roofs has the 5, and a cell that sees both roofs sees it.
Turn the picture onto columns and the roof runs sideways. Two columns share a row, and the free ends sit in different rows. Same argument. I still find the row version first, and the column version only after I have talked myself out of a rectangle.
What I actually scan
- Pick a digit that is down to two cells in a row, or in a column.
- Find a parallel line with two cells of that digit, sharing exactly one crossing line.
- The other two cells are the roof. If they share a crossing line as well, stop. That is an X-wing, and it clears a whole line.
- Any other cell that sees both roof cells loses the digit. The four pattern cells keep it.
Rows 3 and 8, the digit 5
Each of those rows has two 5s, and column 3 is the end that lines up. Step through. The 5 leaves two cells that see both roofs, and one of those cells becomes 1.
See the pattern
Look at the 5s in these two rows.
The four cells, then the two that pay
Row 3 is nearly placed. Row 3, column 1 is 3. Row 3, column 2 is 9. Row 3, column 4 is 2, row 3, column 5 is 4, row 3, column 8 is 6, and row 3, column 9 is 8. Three cells are still open. Row 3, column 3 can be 5 or 7. Row 3, column 6 can be 1 or 7, and it has no 5. Row 3, column 7 can be 1 or 5. Two homes: column 3 and column 7.
Row 8, column 1 is 1. Row 8, column 4 is 4 and row 8, column 7 is 9. The other open cells are row 8, column 2 (2, 7, or 8), row 8, column 5 (3 or 6), row 8, column 6 (3 or 6), and row 8, column 8 (2 or 8). None of them lists a 5. The 5s are row 8, column 3 (2, 5, or 7) and row 8, column 9 (2, 5, or 7).
The base is column 3. The roof is row 3, column 7 and row 8, column 9. Row 2, column 9 can be 1, 3, or 5. It shares box 3 with row 3, column 7, and it shares column 9 with row 8, column 9. The 5 leaves. 1 and 3 stay, and the cell is not solved.
Row 7, column 7 can be 1 or 5. It shares column 7 with row 3, column 7, and it shares box 9 with row 8, column 9. The 5 leaves, and 1 is the only mark left. The step places 1 in row 7, column 7.
One roof is a near miss
The look-alike is the X-wing you almost had. Both free ends in one column, and you clear that column. Here column 7 and column 9 are different, so the deletion is not a whole line. Row 7, column 9 still lists 1, 4, and 5. It sees row 8, column 9, and it does not see row 3, column 7. One roof is not enough. The 5 stays.
The other miss is cleaning the base. Row 2, column 3 still lists 2, 5, and 6. It sits in the shared column, beside the pattern, and it does not see both roofs. I have deleted that 5 because the column looked used. The column is why the roofs work. The digit leaves the cells that see the roof, and the base cells keep their candidates.
Just after the rectangle
Skyscrapers are Expert. On this site they sit after the X-wing and before the two-string kite. A puzzle that needs one is not Mental. The 1,483 archived NYT puzzles never need a skyscraper. Their hardest steps stop at triples.
I look for the rectangle first. When the fourth corner has slid into the next column, and each line really has only two cells, this is the step. A pointing pair inside one box is still cheaper when it is on the board. I take that before I go hunting roofs.
That jumps a puzzle forward until Skyscraper is the next move. Or start a full Expert puzzle.