How to solve skyscrapers puzzles

A skyscrapers puzzle (also called towers or skyline) looks like Sudoku with numbers around the outside. This guide covers the rules, the six deductions that solve almost every board, and one complete worked example you can check with the free solver. Every Vantage daily puzzle can be solved with these ideas alone, with no guessing.

The rules in one minute

The grid is N by N. Put a height from 1 to N in every cell so that each row and each column contains every height exactly once, like a Latin square. The numbers outside the grid are clues: each says how many buildings you can see when you look along that row or column from that edge. A building hides every shorter building behind it, so you see a building only if it is taller than all of the buildings between it and you. Looking along 2 1 4 3 from the left you see the 2 and the 4 (a clue of 2), and from the right you see the 3 and the 4 (also 2). A missing clue gives no information from that side. Some boards also give a few cells for free.

Six deductions that do most of the work

1. A clue of 1 puts the tallest building at the edge

If you can only see one building, it must be the tallest, and it must be right next to you. A 1 at the end of a line of 5 means the 5 stands in the cell next to that clue.

2. A clue equal to N makes the line climb

If you can see all N buildings, they must rise in order: 1, 2, 3, up to N, moving away from the clue. In the worked example below, row 3 carries a 5 on its left, so it reads 1 2 3 4 5.

3. A clue of 2 keeps the tallest building out of the edge cell

The tallest building at the edge would hide everything, giving a clue of 1. So with a 2, the edge cell is never N. (We checked this and every rule on this page against all orderings of 4, 5 and 6 buildings with a short program.)

4. The height cap: a clue limits how tall early cells can be

With a clue of c on an N-long line, the cell k steps from that edge (counting the edge cell as 1) can never hold a height above N - c + k. A 4 clue on a 5-line says the edge cell is at most 2, the next at most 3, then 4, then 5. Tall buildings can only appear late when many must be visible.

5. Two clues from opposite ends

The two clues on one line always add up to at most N + 1, because the tallest building is seen from both ends and every other visible building is on one side or the other. When they add up to exactly N + 1, every building on both sides of the tallest one is visible, so the tallest one sits exactly as many steps in as the first clue says: clues 2 and 4 on a line of 5 put the 5 in the second cell from the clue that says 2.

6. Latin-square elimination

Whenever you place a height, remove it from the rest of its row and column. Write candidates as pencil notes (the Notes button on the daily puzzle) and look for a cell with one candidate left, or a height with only one cell left in its line.

A complete worked example

This is a real 5 by 5 board (practice board 3 in Vantage, shown here free as an example), with 11 clues and one given cell (the 4 in row 2, column 3). The clues run around the edge: top, bottom, left and right.

The solver found it by the line-by-line reasoning below, and it has exactly one answer (checked by an exhaustive search, which counted 1). Here are the first six deductions, in the solver's words:

1. Row 1 (2 from the right): of the 50 orders of 1 to 5 that fit these clues and the cells already known, row 1 column 4 can only be 1, 2, 3 or 5; row 1 column 5 can only be 1, 2, 3 or 4.
2. Row 2 (2 from the left and 3 from the right): of the 3 orders of 1 to 5 that fit these clues and the cells already known, row 2 column 1 can only be 1, 2 or 3; row 2 column 2 must be 5; row 2 column 4 can only be 1 or 2; row 2 column 5 can only be 2 or 3.
3. Row 3 (5 from the left and 1 from the right): only one order of 1 to 5 fits these clues and the cells already known, so row 3 column 1 must be 1; row 3 column 2 must be 2; row 3 column 3 must be 3; row 3 column 4 must be 4; row 3 column 5 must be 5.
4. Row 4 (2 from the left): of the 50 orders of 1 to 5 that fit these clues and the cells already known, row 4 column 1 can only be 1, 2, 3 or 4; row 4 column 2 can only be 1, 2, 3 or 5.
5. Row 5 (1 from the left): of the 24 orders of 1 to 5 that fit these clues and the cells already known, row 5 column 1 must be 5; row 5 column 2 can only be 1, 2, 3 or 4; row 5 column 3 can only be 1, 2, 3 or 4; row 5 column 4 can only be 1, 2, 3 or 4; row 5 column 5 can only be 1, 2, 3 or 4.
6. Column 1 (3 from the top and 1 from the bottom): only one order of 1 to 5 fits these clues and the cells already known, so row 1 column 1 must be 3; row 2 column 1 must be 2; row 4 column 1 must be 4.

Notice how the early steps are the rules above at work: row 3 climbs because of its 5, and row 5 needs the 5 first because of its clue of 1. The finished board:

You can paste these clues into the skyscrapers puzzle solver (press Load an example, then Explain the steps) and see the same list. The solver works with exhaustive lists of orders for a line, which is more than a person can hold in their head; the six deductions above are the human version of the same reasoning.

Common mistakes

• Reading a clue from the wrong edge. Right and bottom clues look back along the line towards you, so read the line in reverse order.
• Counting a hidden building. Only buildings taller than everything before them count, so keep a running maximum as you scan.
• Forgetting the Latin-square rule while chasing the visibility rule. Both must hold at once.
• Guessing. A well-made board never needs it; when you feel stuck, write candidates in every cell and apply deductions 4 and 5 to each line again.

Practise with a fresh board

Vantage gives you one new skyline puzzle a day, free, with no ads and no account: play today's puzzle. Mondays are a gentle 4 by 4 and the weekend boards are 6 by 6 with fewer clues.