Skyscrapers puzzle: rules and how to solve it

In the skyscrapers puzzle, each cell holds a building of height 1 to N, and the clues around the edge tell you how many buildings you can see from that side. Play the skyscrapers puzzle online free here, on grids from 5×5 to 7×7.

Taller buildings hide shorter ones behind them, so a clue is really a statement about how heights rise along a line. Learning how to solve a skyscrapers puzzle is mostly learning to read those clues.

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The rules

The grid is N×N (5×5 to 7×7 here). Each row and column contains the heights 1 to N exactly once.
Each digit is a building with that many storeys.
A clue outside the grid is the number of buildings visible looking along that row or column from that side. A building is visible if it is taller than every building in front of it.
An edge without a clue gives no information.

How to start

Clues of 1 and N

A clue of 1 means the tallest building, N, stands right next to it. A clue of N means you can see every building, so the line climbs 1, 2, 3 … N from that side.

The distance rule

The cell k steps in from a clue c can be at most N − c + k. The reason: if the building there has height h, at most k buildings are visible up to it and at most N − h taller ones beyond it, so c ≤ k + N − h.

From the left, 4 visible keeps the 5 out of the first three cells; from the right, 2 visible keeps it out of the nearest cell. So the 5 is fourth. The last cell is hidden from the left, so the first three must all be visible: they climb.

Opposite clues

Both ends of a line can see the tallest building, so the two clues on a line add up to at most N + 1. A pair that adds to exactly N + 1 means every building is visible from one side or the other.

Skyscrapers is a Latin square underneath, so all the row-and-column logic in our solving techniques guide applies too.

Common mistakes

Counting a building that is hidden behind a taller one.
Forgetting that the tallest building is always visible from both ends.
Reading a clue along the wrong line. Each clue looks straight across its own row or column.

How the ranks scale difficulty

The same four ranks apply here as everywhere on the site. Lemma is a gentle warm-up, Proposition needs a real idea or two, Theorem is near-minimal with almost every clue load-bearing, and Conjecture is the sparsest of all, so expect to think hard.

Skyscrapers grids run from 5×5 to 7×7. Bigger grids allow longer chains of visibility, so higher clues become more subtle.

There are 99,999 puzzles per rank, each with its own number, so you can replay one or share it. Every type also has a 30-puzzle collection that climbs from Lemma to Conjecture. Every puzzle is checked to have exactly one solution that you can reach by logic alone, with no guessing.

Frequently asked questions

How do you solve a skyscrapers puzzle?

Start with clues of 1 and N, use the distance rule to cap every cell near a clue, place the tallest building in each line, then fill in using ordinary Latin-square logic.

What does a clue of 1 mean?

The tallest building is in the cell right next to the clue.

Can opposite clues add up to more than N + 1?

No. The tallest building is visible from both ends, so the two counts overlap by at least one.