How to solve Skyscrapers: seeing the city from the edges
5 min read ·
In Skyscrapers, every cell is a building between 1 and n storeys tall, and every row and column has one building of each height. The numbers around the edge say how many buildings you can see from that side: a taller building hides every shorter one behind it.
The clues look mysterious at first, but each one tells you something exact. This guide goes from the easiest clues to the subtle ones.
The rules
- On an n×n grid, every row and column has one building of each height from 1 to n.
- A clue outside the grid counts the buildings visible from there, looking along that row or column.
- A building is visible if it's taller than every building in front of it.
- If the clues alone would allow two answers, a few heights are given so the solution is unique.
The easy clues: 1 and n
- A clue of 1 means you see only one building, so the tallest (n) stands right next to the clue.
- A clue of n means you see every building, so the line rises 1, 2, 3 … n from that side.
- A clue of 2 means the tallest can't be first (you'd see just 1), and the first building is the only one in front of it that you see.
How close can the tallest building be?
Every visible building before the tallest is a step up, so with a clue of c, the tallest building (n) can't be in the first c − 1 cells from that side. On a 5×5 grid, a clue of 4 puts the 5 in the last two cells of the line.
The same idea limits other cells. With a clue of c, the first cell from that side can be at most n − c + 1: there must be room for c − 1 taller buildings after it.
Opposite clues work together
Every line has two clues, one at each end, and the tallest building is visible from both. So the two clues together tell you roughly where it stands:
- Two big clues on opposite ends push the tallest building towards the middle.
- A small clue at one end and a big one at the other put the tallest building near the small clue.
- On any line, the two clues add up to at most n + 1.
Then think like Sudoku
Once some buildings are placed, the puzzle becomes a Latin square: each height appears once per row and column, so singles and pairs finish the grid. Check every clue again whenever a line is almost full: the last cells usually have only one order that gives the right count.
A solving routine
- Fill every clue of 1 (the tallest next to it) and every clue of n (rising order).
- For each other clue, cross the tallest building out of the cells too close to it.
- Compare opposite clues to locate the tallest building on each line.
- Use row and column logic: each height once per line.
- Count visible buildings on nearly full lines to place the last cells.
Practise
Puzzimo's Skyscrapers comes in 4×4, 5×5 and 6×6, with clues on all four sides; the daily challenge is 5×5. Notes, undo and hints are built in (a hint adds 30 seconds to a ranked time), and a wrong height counts as a mistake; the third ends the round.
Frequently asked questions
How do I count visible skyscrapers?
Look along the line from the clue. The first building is always visible; every later building is visible only if it is taller than all the buildings before it.
What does a clue of 2 mean in Skyscrapers?
You see exactly two buildings: the first one, and the tallest further along. So the tallest can't stand right next to the clue, and every building between the first and the tallest is shorter than the first.
Where should I start a Skyscrapers puzzle?
With the clues equal to 1 or to the grid size, which place buildings immediately. Then look at lines with two large clues, where the tallest building must sit near the middle.