Two towers, one guarantee

The short version

A skyscraper is two parallel lines — say two rows — where a digit has exactly two homes in each. One home in each row lies in the same column: that is the base. The two remaining homes are the towers.

Only one cell in the base column can hold the digit. Whichever one it is, the other row is forced to place its digit at its tower. So at least one tower holds the digit, and anything seeing both towers cannot.

The shape

Two rows, digit D, two homes each:

• Row A: homes at C1 and C2.
• Row B: homes at C1 and C3 — note the shared column C1, and the different columns C2 and C3.
• C1 in both rows is the base.
• The cells at C2 and C3 are the towers.

Because C1 holds at most one digit, at most one base cell is D. The row whose base cell is not D must place D at its tower. And if neither base cell is D, then both rows place D at their towers.

In every case: at least one tower is D. Never both, and never neither.

Eliminate D from any cell that sees both towers.

Why it works

The argument is a short exhaustion, and the key move is realising what the base column cannot do: hold two copies of the digit.

• Base cell in row A is D. Then row B's base cell is not D. Row B needs a D, and its only other home is its tower. So row B's tower is D.
• Base cell in row B is D. Symmetric: row A's tower is D.
• Neither base cell is D. Then both rows need D elsewhere, which means both towers are D.

All three cases produce at least one tower. That is the guarantee.

Note how much weaker this is than an X-Wing, and how much more often it occurs. The X-Wing demands four cells forming a perfect rectangle. The skyscraper only demands one shared line — the two towers are free to sit in any columns at all. Fewer conditions, more patterns, weaker conclusion.

A real board

The digit is 2.

• In row 1, 2 can only go in R1C8 or R1C9.
• In row 8, 2 can only go in R8C7 or R8C9.
• Both pairs include column 9: R1C9 and R8C9 form the base.
• The towers are R1C8 and R8C7.

At most one of R1C9 and R8C9 holds a 2. So at least one of the two towers holds one:

• If R1C9 is 2, then R8C9 is not, so row 8 must place its 2 at R8C7.
• If R8C9 is 2, then R1C9 is not, so row 1 must place its 2 at R1C8.
• If neither is 2, then both rows place their 2 at the towers.

Now look at R7C8, which holds {1, 2, 6, 7, 9}. It shares column 8 with the tower R1C8, and it shares the bottom-right box with R8C7. So it sees both towers. Since at least one tower is 2, R7C8 cannot be 2.

One elimination, and the cell is still not solved — but its candidate list just got shorter, and that is how hard boards move.

How to scan for one

Skyscrapers are found by digit, like every fish-family pattern, and they are faster to scan than X-Wings because the condition is looser.

1. For each digit, list every line with exactly two homes. Both rows and columns. Lines with one home are singles; lines with three or more are not usable here.
2. Compare pairs of parallel lines. Two rows with two homes each — do they share a column? If yes, the two remaining homes are towers.
3. Confirm the towers differ. If two rows both have homes in the same two columns, that is an X-Wing, which is stronger. The skyscraper case is when they share exactly one column.
4. Hunt the eliminations. For each cell containing the digit, check whether it sees both towers — same column as one, same row or box as the other.
5. Do the column pass too. The pattern is symmetric. Two columns sharing a row produce eliminations along rows, and solvers who only scan rows miss half of them.

The productive case is a digit that is nearly finished. With only two homes per line, the digit is close to completion, and those are exactly the lines that form skyscrapers.

Skyscraper vs X-Wing

The X-Wing is the stronger pattern — four eliminations instead of one or two — but it occurs less often. The skyscraper is the everyday version, and on a hard board you will usually find three skyscrapers for every X-Wing.

There is also a two-string kite and an empty rectangle in this family, both of which are variations on the same "one cell must hold the digit, so these two cannot both be it" reasoning. If the skyscraper makes sense, those are not new ideas.

The traps

Lines that share both columns. That is an X-Wing, not a skyscraper. Applying skyscraper logic still gives a valid but weaker elimination, so you will not break the board — you will just miss the stronger move.

Eliminating from a cell that sees one tower. The guarantee is "at least one tower", not "both". A cell that sees only R1C8 is unconstrained: the 2 might be at R8C7. Both towers must be in view.

Forgetting the box direction. Towers can be seen through a box as well as through a row or column. In the board above, R7C8 sees R8C7 through their shared box, not through a line. Solvers who only check rows and columns miss these eliminations entirely.

Three homes misread as two. A line with three homes for the digit cannot be half of a skyscraper, even if two of them share a column with another line. The pattern needs exactly two.

Where to use it

Skyscrapers belong in the same band as XY-Wings, and they are usually the first thing to check when the fish scan comes up empty. They cost less than an XY-Wing to find — you are comparing line positions, not candidate lists — and they fire more often.

If a digit has two homes in three or four different lines, that digit is a skyscraper factory. Sweep it before moving on to anything more exotic.