Skyscrapers

Skyscrapers is a Latin-square logic puzzle on an n×n grid where each cell holds a building of height 1 to n, every height appears once per row and once per column, and edge numbers tell you how many buildings are visible from that side. It is also published as Paper Skyscrapers or Towers — the same puzzle under three names.

What does each height and row mean?

The puzzle is played on a square grid where every cell holds a building of a certain height. On an n×n grid the heights run from 1 to n, and, exactly as in a Latin square, each height must appear once in every row and once in every column.

How do the edge clues work?

What makes it Skyscrapers are the clues around the edge. The number beside a row or column tells you how many buildings are visible looking along it from that side. A taller building hides every shorter building behind it, so a clue of 1 means the tallest building (n) stands right at that edge, while a clue of n means the heights climb 1, 2, 3 … in order away from you.

Together the Latin-square rule and the visibility clues pin down a single answer. Every board on this page is generated and checked to have exactly one solution reachable by logic alone — no guessing, just deduction about which buildings can be seen from where.

To solve Skyscrapers, read the edge clues like stories about height — a 1 forces the tallest building at the edge, a full-size clue forces a staircase — then let the Latin-square rule eliminate the rest, never guessing. The core techniques:

• A clue of 1. If a side shows 1, the cell nearest that edge must be the tallest building, n — it hides everything behind it.
• A clue of n. A clue equal to the grid size forces a full staircase: 1, 2, 3 … n in order from that edge.
• High clues push the tall buildings back. The bigger the clue, the further from the edge the tallest building must sit, because several buildings have to be visible first. A clue of 2 usually keeps n out of the front cell.
• Combine with Latin-square logic. Once a few heights are placed, the “each height once per row and column” rule eliminates candidates exactly as in Sudoku.
• Pair the opposite clues. The clues at the two ends of the same line constrain it together; a low-and-high pair often fixes where the tallest and shortest go.

Work from the most telling clues first — the 1s and the full-size clues — then let the Latin-square eliminations cascade. Never guess: a proper Skyscrapers always has a forced height somewhere on the grid.

On a 4×4 board, a 1 clue on the left of a row forces the tallest building (4) into the first cell, and a 2 clue on the right then places the 3 — from there the column clues and the once-per-row rule resolve the grid one forced cell at a time.

Starting from a 1 clue

Consider a 4×4 board with a 1 clue on the left of a row. A 1 means only one building is visible from the left, which can happen only if the tallest, 4, sits in the very first cell — it blocks the other three. Write the 4 in immediately.

Combining the opposite clue

Now suppose the right-hand clue on that same row is 2. Two buildings are visible from the right, so the second tallest seen from that side must be the 3, placed so that exactly one taller building (the 4) hides the rest. Combined with the 4 already fixed on the left, the row’s heights quickly fall into place.

From there the column clues and the “each height once” rule take over: every height you place removes a candidate from its row and column, and the grid resolves one forced cell at a time — never a guess.

Penciling candidates as you go

It helps to keep a running note of which heights each cell could still take, exactly as you might pencil candidates in Sudoku. When a clue or a placed building rules a height out of a cell, cross it off; when a height has only one possible home left in a row or column, write it in. Alternating between the visibility clues and this candidate elimination is what carries you from the first forced corner to a finished skyline — on the 5×5 board especially, the clues open the grid and the Latin-square logic closes it.

This page offers Skyscrapers at 4×4 and 5×5 — the 4×4 is a gentle introduction, while 5×5 adds a height and more interplay between the visibility clues and the Latin-square logic. The 4×4 grid has few heights, and clues that often force a corner straight away, whereas the 5×5 board noticeably ramps up the deduction.

The number of edge clues also shapes difficulty: a board with clues on all four sides hands you many footholds, whereas a sparser board makes you lean harder on row-and-column elimination. Tap New for a fresh board at either size, or play the shared Daily to solve the same skyline as everyone else that day.

Skyscrapers, Paper Skyscrapers and Towers are three names for the same puzzle — a Latin square read through visibility clues — making it kin to the other number-placement puzzles in this collection.

The puzzle goes by several names: Skyscrapers, Paper Skyscrapers — the pen-and-paper version, to distinguish it from look-alike apps — and Towers. They are the same puzzle: a Latin square read through visibility clues.

At heart Skyscrapers is a Latin-square puzzle, which makes it kin to the number-placement puzzles in this collection. If you enjoy filling a grid so each value appears once per row and column, try Futoshiki, which adds greater-than and less-than signs, or Sudoku, which adds 3×3 boxes. The deductive skill transfers directly.

The key Skyscrapers terms are height (a cell value 1 to n), clue (an edge number of visible buildings), visible building (one taller than everything in front of it), and Latin square (each height once per row and column). A few words help when reading guides:

• Height — the value 1 to n in a cell, standing for a building’s height.
• Clue — an edge number giving how many buildings are visible from that side.
• Visible building — one taller than everything in front of it; shorter buildings behind a taller one are hidden.
• Latin square — a grid in which each height appears exactly once in every row and column.

You don’t need the terms to play — just tap a cell to cycle its height — but they make strategy easier to follow.

The most common Skyscrapers mistakes are forgetting the Latin-square rule, misreading clue direction, guessing heights you cannot justify, and ignoring the opposite clue on a line. Most errors come from a few habits:

• Forgetting the Latin-square rule. The visibility clues grab attention, but each height must still appear once per row and column — break that and the board is wrong.
• Misreading direction. A clue counts buildings visible from its own side; read it looking inward from that edge, not the other way.
• Guessing heights. A height you cannot justify can satisfy one clue while breaking another. Find the forced cell instead.
• Ignoring the opposite clue. The two clues on a line work together; using only one leaves easy deductions on the table.

The strongest Skyscrapers deductions come from reading the two clues on the same line together — because they describe one row or column from opposite ends, a pair often pins down far more than either clue alone.

How do specific clue pairs constrain a line?

• 1 and n. A 1 at one end fixes the tallest building there; an n at the other end forces a full staircase — and the two can only agree in a single arrangement.
• A high clue plus a low clue. The high side wants its tall buildings spread out and pushed back; the low side wants a tall building near its edge. Reconciling the two usually fixes the extremes of the line.
• Clues share a budget. Two clues on one line can never both be large at once — there is only so much visibility to go round — so a big number on one side quietly caps what the other can be.

Why glance at the opposite clue first?

Getting into the habit of glancing at the opposite clue before you commit a height turns many “hard” lines into forced ones, and it works hand in hand with the row-and-column elimination you already use. On a 5×5 board especially, clue pairs are often the key that unlocks the first full line.

Skyscrapers turns a dry Latin square into a skyline you can almost see — a visual hook layered over genuine logical depth, fair and luck-free, that sharpens skills carrying straight over to Sudoku and Futoshiki.

That visual hook is what makes it so moreish: a little skyline that has to look a certain way from each side. A 4×4 is a brisk warm-up; a 5×5 is a meatier solve of the kind competition setters love, and a regular sight at puzzle championships.

It is fair and luck-free — every board has one answer reachable by reasoning — so the shared Daily is an enjoyable thing to compare with friends. And because it is built on the same one-of-each-per-row-and-column rule as the number puzzles here, it sharpens skills that carry straight over to Sudoku and Futoshiki, while adding a spatial, sightline-reading twist all of its own.

Skyscrapers belongs to the family of Latin-square logic puzzles — the same lineage as Sudoku and Futoshiki — and grew up in puzzle magazines and competitions rather than having a single inventor, remaining a regular fixture at events such as the World Puzzle Championship.

It is published under a handful of names — Skyscrapers, Towers and Paper Skyscrapers — but the idea is always the same: arrange the heights so that what you can see from each edge matches the clue. It pairs a simple, visual story with deep logical structure, which is exactly why it has endured.