What is the Skyscrapers puzzle?
Skyscrapers is a logic puzzle on an N×N grid: build a tower of height 1 to N on every lot so that no height repeats in a row or column, and so that each number on the edge equals how many towers are visible from that side. It is also called the towers puzzle or building heights puzzle, and it belongs to the same Latin square family as sudoku.
Everything rests on one fact: a taller tower hides the shorter towers behind it. A person standing at the edge sees one more tower every time a building rises above everything in front of it. Looking down the heights 2, 4, 1, 3 they see the 2 and the 4 only, because the 1 and the 3 are hidden behind the 4, so their clue is 2. In this game every number you enter raises a real tower on the lot, so you can see at a glance which towers block which.
What are the exact rules?
There are three. No repeated height in a row, no repeated height in a column, and each edge clue must match the number of towers visible from that side. Each board has exactly one arrangement that satisfies all three.
| Rule | Meaning |
|---|---|
| Row | Each row holds heights 1 to N once. On a 5×5 board that is one tower each of 1, 2, 3, 4 and 5 floors |
| Column | Each column also holds 1 to N once |
| Clue | The number beside a person is how many towers they see looking along their row or column. A tower is visible only if it is taller than everything in front of it |
| Given towers | Grey towers stand from the start and cannot be demolished. They appear only on the larger boards |
| Finish | Every lot built and no row or clue breaking a rule. The sun sets and the windows light up |
How do I read the clue numbers?
The most useful clues are 1 and N. A 1 means the lot right in front of that person holds the tallest tower N. An N means the towers rise in order 1, 2, 3, ... N from that side. Every other number still limits what can stand in the first lot.
| Clue | What it tells you (N = 5) |
|---|---|
| 1 | The first lot is 5. Everything behind is hidden, so any order works there |
| 5 (= N) | The line is fixed as 1, 2, 3, 4, 5 from that side |
| 2 | The first lot is not 5. If the first lot is 4 the 5 can be anywhere; if it is 3, the 4 cannot stand in front of the 5 |
| k | The first lot is at most N - k + 1, and the 5 stands in the k-th lot from that side or further back |
| Opposite clues add to N + 1 | With a on the left and b on the right, a + b = 6 fixes the 5 in the a-th lot from the left. For 2 and 4 the second lot is the 5 |
| Opposite clues add to more than N + 1 | Impossible, so it never appears on a board |
How do I solve it?
- Handle the 1 and N clues first. An N goes in front of a 1, and 1, 2, 3... in front of an N. Several lots settle at once.
- Place the tallest tower early. The N floor tower is visible from every side and hides everything behind it. With a clue k the N can only stand in the k-th lot or further, so pencil the possible lots for N in every row and the rest falls into place.
- Add the two clues of a line. If they sum to N + 1 the position of N is fixed immediately.
- Write the ceiling of the first lot. Next to a clue k the first lot cannot exceed N - k + 1. Beside a 3 on a 5×5 board the first lot is 3 or less.
- Count what is missing in each row and column. Four towers built means the last lot takes the missing height. If a height fits only one lot in its row, that lot takes it even with other candidates pencilled in.
- Stuck? Press Check. Rows that break a rule turn red along with their clues. Hint builds one tower that can be deduced from the current board.
How do the sizes differ?
4×4 is a warm up for learning to read the clues, 5×5 is the classic size, and 6×6 and 7×7 are for experts who pencil in candidates. On the big boards a few towers are given from the start so that every board can still be finished by deduction.
| Size | Clues | Given towers | Target time |
|---|---|---|---|
| Easy 4×4 | About 8 of 16 | 0 to 2 | About a minute |
| Normal 5×5 | About 9 of 20 | 0 to 2 | About 3 minutes |
| Hard 6×6 | About 11 of 24 | 3 to 6 | About 6 minutes |
| Expert 7×7 | About 13 of 28 | 4 to 8 | 10 minutes or more |
Every board can be finished by pure deduction. You never have to guess and hope. The rank on the result screen is based on time, and hints used are listed separately.
How are the boards made?
The game shuffles a random Latin square, counts the visible towers from all four sides to get 4N clues, then removes the clues one at a time. A clue stays removed only if a propagation solver, which scans every possible arrangement of each line to narrow the candidates, can still finish the board; otherwise it goes back. On large boards a few towers are built first so the solver never stalls, and those are thinned the same way. Finally a backtracking solver confirms there is exactly one answer.
That is why boards of the same size have slightly different numbers of clues and given towers. The share link carries the board number, so a friend who opens it gets the very same board to race on.
What is the daily puzzle?
The daily puzzle is a 5×5 board that changes every day and is the same for everyone on that day. It rolls over at midnight Korean time, and finishing it is recorded on your attendance streak.
Best times are stored separately for each size. If you leave in the middle of a board, the Continue button brings you back to the same spot.
Controls
- Phone and mouse: tap a lot or a tower to select it (orange outline), then tap a height button below the board (to the right in landscape) and the tower rises. Tapping the same height again demolishes it. With Memo on, numbers are pencilled onto the lot as candidates.
- Keyboard: digits 1 to N build, Backspace, Delete or 0 demolish, arrow keys move, N toggles memo, H hint, Z undo, C check.
- Buttons: Undo steps back through your entries one at a time. Hints used are shown with your result.
Records and unfinished boards are kept only in this browser and never sent to a server. Clearing browser data clears them too.