Skyscrapers rules
Skyscrapers is a Latin square with a view: fill the grid with heights so that each row and column has every height once, and so that every clue sees the right number of towers.
- Each row and each column contains every height from 1 to N exactly once.
- A clue tells how many towers are visible from that side, looking along the row or column; taller towers hide shorter ones.
- Lines without a clue can show any number.
Skyscrapers is also known as towers.
Try it on small grids
Three small Skyscrapers puzzles to get the feel. Hints work here too and explain each step.
Practice 1 · 4x4 · Easy
Practice 2 · 4x4 · Easy
Practice 3 · 5x5 · Medium
Your first moves
Find the 1s and the Ns first. A clue of 1 puts the tallest tower right next to it. A clue of N fills its whole line, 1 to N from that side. Next, a big clue keeps tall towers away from its edge: with a clue of 4 on a 5x5 grid, the first cell can be at most 2. Then use the usual Latin-square logic: a height that fits only one cell of a line goes there.
A worked example, step by step
Our solver finished this Skyscrapers (4x4) in 16 steps. Every step names its technique; step through it on the board, or read the whole solve below.
All 16 steps
- Clues 1 and N. The 4 at the bottom of column B sees every tower, so the heights rise 1 to 4 from that side.
- Naked single. B1 can only be 4, so no other cell of its row or column is 4.
- Naked single. B2 can only be 3, so no other cell of its row or column is 3.
- Naked single. B3 can only be 2, so no other cell of its row or column is 2.
- Naked single. B4 can only be 1, so no other cell of its row or column is 1.
- Distance from the clue. The 3 at the left of row 2 needs 2 shorter towers in front of the tallest it sees, so the first cells cannot be too tall: A2 at most 2.
- Distance from the clue. The 3 at the left of row 4 needs 2 shorter towers in front of the tallest it sees, so the first cells cannot be too tall: A4 at most 2.
- Naked single. A4 can only be 2, so no other cell of its row or column is 2.
- Naked single. A2 can only be 1, so no other cell of its row or column is 1.
- Naked single. A1 can only be 3, so no other cell of its row or column is 3.
- Naked single. A3 can only be 4, so no other cell of its row or column is 4.
- Line permutations. Only 1 order of row 4 fits its candidates and clues; C4, D4 lose the heights no order puts there.
- Naked single. C4 can only be 3, so no other cell of its row or column is 3.
- Naked single. C3 can only be 1, so no other cell of its row or column is 1.
- Naked single. C1 can only be 2, so no other cell of its row or column is 2.
- Naked single. C2 can only be 4, so no other cell of its row or column is 4.
Common mistakes
- Counting towers that are hidden: a 2 behind a 4 is not visible.
- Reading a clue on the right as if you looked from the left: every clue looks into the grid from its own side.
- Forgetting the clues at both ends of the same line: they constrain it together.
Where Skyscrapers comes from
Skyscrapers has been a regular in puzzle magazines and logic championships for decades, and is also published as Towers. Its rules combine a Latin square with a line-of-sight count.