Skyscrapers Puzzle

Skyscrapers is a Latin square with clues around the edge. Each cell holds a skyscraper whose height is a number from 1 to n, every row and column holds each height once, and the numbers outside the grid say how many skyscrapers you can see from that side. You can play grids from 4×4 up to 6×6, and every puzzle here has exactly one solution that you can reach by logic, with no guessing.

Rules

An n×n grid takes the heights 1 to n. Each height appears exactly once in every row and exactly once in every column.

A number outside the grid looks along its row or column from that side. It counts the skyscrapers you can see: a skyscraper is seen when it is taller than every one in front of it, and a taller one hides every shorter one behind it. A clue of 1 sees only the tallest, so the tallest stands right next to it. A clue of n sees every height, so the heights rise 1, 2, 3 and on from that side.

Some edge slots are empty. An empty slot has no clue, and that side of the line can see any number. No cell starts with a digit.

The puzzle is solved when every row and column has each height once and every clue sees exactly its count.

Controls

Tap a cell to select it. Its row and column are tinted, the clues at both ends of them light up in violet, and every cell with the same height is shaded a little darker. Tap a height on the keypad under the board, or type it. Erase, Backspace, Delete or 0 clears the cell, and the arrow keys move around the grid. The small number on each keypad key counts how many of that height are still to place.

Notes are small pencil marks for the heights a cell could still be. Turn on Notes with the button or the N key, then each height you tap or type switches that mark on or off. Shift with a digit adds or removes a note without turning Notes on. When you place a height, that height’s notes disappear from the rest of its row and column, in the same undo step.

A height that repeats in a row or column turns red as soon as it appears. When a full row or column misses one of its clues, that clue turns red and its line is tinted. A clue whose full line sees the right count turns soft teal, and so does a finished row or column. Press Hint for one step the rules already force, with a sentence saying why; the cell and the clues it uses light up. Press Check when you want the board checked against the rules. Undo and Redo step through your moves, and Reset clears the grid.

Techniques

Easy grids follow from single clues and singles:

• Clues of 1 and n: a 1 puts the tallest height next to it. An n fills its whole line from 1 up to n.
• Too close to a clue: a clue of c cannot have a tall height right next to it. The first cell holds at most n − c + 1, and each step in allows one more.
• Facing clues: when the clues at both ends of a line add up to n + 1, the tallest height stands as many cells in from one end as that end’s clue.
• Singles: a cell with one height left takes it, and a height with one place left in a row or column goes there.

Medium grids also need:

• The second tallest: next to a clue of 2, height n − 1 stands either in the first cell or somewhere behind the tallest height.
• Pairs: two cells in a line that can only be the same two heights take those heights away from the rest of the line. Two heights that fit only two cells clear the other notes in them.

Hard grids add triples, a look at whole lines, and a short try:

• Line orders: list the orders of heights a line can still have under the clues at both ends. A height that no order allows in a cell comes off.
• One-step try: when a cell has two heights left and one of them breaks a rule, the cell takes the other.

Beginner tips

Start with the 1s and the largest clues. A 1 places the tallest height at once, and a clue of n fills its whole line.

Next, cap the cells near each clue. A 3 next to a 5×5 row means the first cell is at most 3 and the second at most 4. Write those limits in as notes and look for a height with only one place left.

Remember that a clue only counts what it can see from its own side. The same line read from the other end often has a different count. If you are new to Skyscrapers, start with a 4×4, and move to the daily 5×5 easy grid once the clues of 1 and n feel natural.