Skyscrapers puzzle strategy: how to solve every Rooftops board

Technique 1 · basic

The 1 clue

A 1 on the edge means you see exactly one rooftop from there. The first building always shows, so it has to hide everything behind it, and only the tallest height can do that. Write the tallest building, a 5 on a 5×5 board, in the tile right next to every 1. Nothing else is needed: no notes, no counting. Do it before anything else, because each of these buildings also takes the 5 out of its whole row and column, and that is often what the next step needs.

A 1 says nothing about the rest of its line. The tiles behind the tallest building are hidden whatever they hold, so the line is still open. Its other end may have a useful clue of its own, though, and the far tiles are where that clue does its work.

Where to look: every side of a fresh board. A puzzle usually has two or three 1s, and they are the free start.

Technique 2 · basic

The full clue

The opposite of a 1 is a clue equal to the board size: a 5 on a 5×5 board. Seeing all five rooftops means no building may hide another, so each one must be taller than every building in front of it. There is only one order that does that: 1, 2, 3, 4, 5, rising away from the clue. Fill the whole line at once. A full clue is rarer than a 1, and it is the single most generous clue a puzzle can give, because it settles five tiles and crosses five heights off five other lines.

When a full clue and a 1 face each other across the same line, they agree: the 1 sees the 5 at its end of the line, exactly where the staircase ends. Spotting that pair is a quick check that you read both clues right.

Where to look: any clue equal to the board size. Then check the opposite end of that line for a 1.

Technique 3 · basic

How far the tallest stands from a clue

Every clue between 2 and N - 1 keeps the tall buildings away from it. A clue of c needs c rooftops in view, and every building in view stands in front of the tallest one, so the tallest has at least c - 1 tiles before it. On a 5×5 board a 4 pushes the 5 at least three tiles in. The same count limits each tile: the tile next to clue c is at most N - c + 1, and each step further in allows one story more. With a 4 on a 5×5 board that gives 1 or 2, then up to 3, then up to 4.

Write these limits as notes on the first pass over the edges, clue by clue. A 2 only rules out the tallest building next to it, but a 3 or a 4 fills whole stretches of a line with small notes, and the later techniques work by narrowing exactly those notes.

Where to look: the highest clues that are not N, and the tiles nearest to them.

Technique 4 · basic

Last place and last height

Rooftops is also a Latin square: each height once in every row and column, as in a sudoku without boxes. That gives two quick checks. A hidden single: a height that can go in only one tile of a line goes there. A naked single: a tile that has only one height left goes there. The hidden single is the one players miss, because the tile itself may still look open. Pick a height, often the tallest, and for each row ask where it can still go. Every building of that height already on the board blocks its whole column.

These checks are cheap, so run them after every new building. A clue step on the edge feeds a single in the middle, the single takes a height out of a crossing line, and that line often gives up its next tile by itself.

Where to look: the height with the most copies already on the board. Its last missing places are the easiest to find.

Technique 5 · intermediate

Opposite clues

Two clues at the two ends of one line work together. The tallest building is seen from both sides, so it is counted by both clues, and each clue also needs its other rooftops in front of it. The two clues together can never add up to more than N + 1. When they add up to exactly N + 1, there is no slack at all: the tallest building stands c tiles in from clue c, on both sides at once. On a 5×5 board, a 4 facing a 2 puts the 5 fourth from the 4. A 3 facing a 3 puts it in the middle.

When the sum is lower, the pair still leaves a short window for the tallest: at least a - 1 tiles from one end and b - 1 from the other. Combine that window with the singles: a 5 already in a crossing column often closes the last gap.

Where to look: lines with clues at both ends. Add them up and compare with N + 1.

Technique 6 · intermediate

A 2 and the tallest building

Once the tallest building of a line is placed, a 2 clue becomes very exact. The 2 sees the first building and the tallest, and nothing else. So the first building, the one right next to the clue, must be taller than every building between it and the tallest. Unless the tallest stands right behind it, the first tile is never the lowest height left, and the second-tallest building left is either in that first tile or hidden behind the tallest. When only two or three tiles are open, this usually settles all of them.

Read it together with the columns. In the example, the first tile from the right cannot be the 4 because its column already has one, so it has to be the 3, and the tile in front of it has to be lower still. A 2 with the tallest far away is one of the strongest clues on a half-solved board.

Where to look: a 2 whose line already has the tallest building in place, with a few tiles open between them.

Technique 7 · intermediate

Ruling out heights by visibility

This is the general method behind the last three techniques. Take a clued line with a few tiles open, and test each note in each tile: if that height stood there, could the line still show the right number of rooftops? If no order of the remaining heights works, cross the note out. In practice you rarely need to list all the orders. Find the buildings that will be visible whatever happens, usually the tallest and the second tallest, and count them first. When they already reach the clue, every building in front of them has to stay hidden.

Hidden means lower than something in front of it. That one idea removes most notes: a tall building between the clue and the tallest is visible unless an even taller one stands before it. When both ends of the line have clues, test every note against both.

Where to look: a clued line with three or four tiles open, after you have placed its tallest building or narrowed it to two tiles.

Technique 8 · advanced

Look-ahead on visibility counts

Hard boards come to a point where no single line gives anything. Then pick a tile with two notes, ideally next to a clue, and imagine one of them. Follow only the forced steps from there: singles and edge clues, nothing that needs a choice. If a clue soon has to see too many or too few rooftops, the imagined height was wrong, and the tile holds the other one. This isn't guessing: you never keep the imagined height, only the proof that it fails, and nothing on the board changes until that proof is complete.

Keep the chain short and watch the clues. Rows with two or three notes per tile run out of options fastest. If a chain gets long without a contradiction, drop it and try the other height of the same tile, or another tile next to a clue.

Where to look: tiles with exactly two notes next to a clue, in lines where a row or column is nearly full.

Common mistakes

• Counting the hidden buildings. A clue counts rooftops you can see, not buildings. A 3 behind a 4 is hidden, however close it stands to the edge.
• Forgetting the far tiles. The edge limits come from one side only. The tiles far from a clue get their limits from the clue at the other end, or from none.
• Reading a 2 as "the tallest is second". A 2 only says two rooftops show. The tallest building can stand anywhere from the second tile to the far end.
• Writing in a guess. Look-ahead happens in your head or in notes. Placing the imagined building and playing on hides the wrong step under later ones.