How to Solve Skyscrapers Puzzles

How the clues work

Take a row of a 5 × 5 grid that reads 2, 4, 1, 5, 3 from the left. Looking from the left you see the 2, then the 4 rises above it, the 1 is hidden behind the 4, the 5 rises above everything, and the 3 is hidden behind the 5. That is three buildings, so the clue on the left is 3. From the right you see the 3, then the 5, and nothing behind the 5 can show: the clue on the right is 2.

A space with no number is simply no clue: that end of the line can see any number of buildings. A few daily grids also give a height or two, drawn in bold; they cannot be changed.

Sizes and difficulty

Every daily grid is checked when it is made: the reasoning in this guide fills it in without any guessing, which also proves that it has exactly one solution.

Clue 1 and clue n

Two clues solve squares on their own. A clue of 1 means only one building is visible, so the first building must be the tallest: write the top height in the square right beside the clue. A clue equal to the size of the grid, n, means every building is visible, which only happens when the heights climb one storey at a time: the line reads 1, 2, 3 and so on up to n from that side.

These are the first things to look for on any day. A clue of 1 places one building, and because each height appears once per row and column, it also tells you where the tallest building is not.

The edge distance rule

The other clues limit how tall the buildings near the edge can be. With a clue of k, at least k − 1 buildings must be seen before the tallest one, so the tallest can't stand in the first k − 1 squares. More generally, the square at position p from the clue (counting the first square as 1) can be at most n − k + p storeys tall.

Write these limits as notes before anything else on medium and hard days. Big clues push the tall buildings away from their edge, and the square beside a big clue is always small.

Clue pairs on one line

The tallest building in a line is seen from both ends, so the two clues of a line share it. If the clue on one end is a and on the other is b, the tallest needs at least a − 1 squares before it on one side and b − 1 on the other, so a + b can never be more than n + 1.

A clue of 2 has a neat rule of its own: if the tallest building is at the far end of the line, the first square holds the tallest of all the others, because nothing between can be seen.

A solving routine

1. Clues of 1 and n first. Place the tallest buildings and the rising lines.
2. Note the edge limits. Use the distance rule to cross out heights near every clue.
3. Find each line's tallest building. Every row and column has one; where can it go, given both clues?
4. Use the Latin square rule. A height with one place left in a row or column goes there, and Notes clear themselves keeps the pencil marks honest.
5. Check full lines. When a row or column is full, its clues get a tick if they are right and turn red if not.

Common mistakes

• Counting hidden buildings. A building behind a taller one is never seen, however tall it is compared with its own neighbours.
• Forgetting the far clue. Every line can have two clues; a row that satisfies the left one may still break the right one.
• Ignoring blank clues. They are not zeros. A blank only means that end of the line gives no help.

Scoring and the leaderboard

Skyscrapers is ranked by time, fastest first; fewer entries break ties. The clock starts with your first move, the grid is checked the moment the last square is filled correctly, and only a member's first solve of the day counts for the leaderboard. Practice puzzles from Play more are never ranked. If you enjoy this kind of reasoning, Futoshiki uses the same Latin square with greater than signs instead of edge clues.