The Skyscraper: an X-Wing with one corner shifted

What a Skyscraper is

Pick a digit. Find two rows in which that digit has exactly two possible cells each. Each of these rows is a strong link: if one cell is not the digit, the other must be. Now suppose one cell from each row lies in the same column. Those two aligned cells are the base of the skyscraper. The other two cells, which do not share a column, are the roofs.

The rule: any cell that sees both roofs cannot hold the digit. The same pattern works with columns instead of rows. Two columns with exactly two candidates each, one end of each sharing a row, give two roofs in different rows.

Why it works

The two base cells share a column, so at most one of them can be the digit. Consider the row whose base cell is not the digit: because that row is a strong link, its roof must be the digit. So in every case at least one of the two roofs holds the digit.

We do not know which roof it is, but a cell that sees both roofs is in the way either way. If that cell were the digit, neither roof could be, and then both base cells would have to be the digit, which the shared column forbids. The same “one of these two” reasoning powers the XY-Wing and longer chains.

How to find a Skyscraper

With complete pencil marks, choose a digit and list every row where it has exactly two candidates, with their columns. Look for two rows that share one column but not the other. If they share both columns, it is an X-Wing instead, which is even better.

Then find the roofs, the two cells that are not aligned, and check the cells that see both of them. Usually these are cells in the same box as one roof and the same line as the other. If any such cell still has the digit as a candidate, it can be removed.

Repeat with columns as the strong links. Because the pattern needs only two strong links on one digit, it is quick to check once you already scan digits for X-Wings; many solvers look for both in the same pass.

A worked example

On the example board, look at digit 1. In row 4 it can only go in R4C3 (row 4, column 3) or R4C8. In row 8 it can only go in R8C2 or R8C8. R4C8 and R8C8 share column 8: they are the base. The roofs are R4C3 and R8C2, which are in different columns, so this is a Skyscraper rather than an X-Wing.

Column 8 can hold only one 1, so at least one base cell is not 1, and the roof in that row must be 1. Either R4C3 or R8C2 is 1. Now R9C3, with candidates 1 and 6, shares column 3 with R4C3 and box 7 with R8C2. It sees both roofs, so it cannot be 1.

With 1 removed, R9C3 is left with only 6, shown in gold. This position comes from a harder puzzle than the ones on Quietdoku. None of the techniques up to Swordfish and XY-Wing make progress here, and the Skyscraper is the step that breaks it open.

Common mistakes

Using lines with more than two candidates. Each of the two rows must have exactly two places for the digit; otherwise it is not a strong link, the roof is not forced, and the elimination is invalid.

Forgetting to check the base. The base cells must be in the same column (or the same row, for column-based skyscrapers). If neither end lines up, nothing links the two rows and there is no pattern. If both ends line up, you have an X-Wing, which removes the digit from both columns instead.

Eliminating from cells that see only one roof. A cell next to one roof can still be the digit when the other roof is the one holding it. Only the overlap of both roofs is safe.

Where the Skyscraper fits

The Skyscraper sits just beyond the X-Wing and Swordfish. It uses the same kind of strong links as fish patterns but joins them like a chain, which makes it a natural bridge to the XY-Wing and to longer chains. You will meet it in very hard puzzles from other sources.

Quietdoku puzzles never require it. Every puzzle has exactly one solution and is graded by the hardest technique a person needs, and Expert puzzles go up to X-Wing, Swordfish and XY-Wing. Knowing the Skyscraper can still give you a shortcut when you spot it early.