Skyscrapers Puzzle Rules: A Checked 4x4 Example

How do you solve a Skyscrapers puzzle?

Fill a Skyscrapers grid so each row and column contains every allowed height once, while each outside clue matches the number of buildings visible from its direction. Taller buildings hide shorter ones behind them. Start with clues that force an entire line or its tallest building, then use crossing rows and columns. The original practice puzzle below has one checked solution.

The rules follow Conceptis's classic Skyscrapers description. Our example uses heights 1, 2, 3 and 4 in a four-by-four grid. There are no separate box rules or diagonal requirements. Each number represents a building's height, not a quantity to add to the edge clue.

What does an outside clue count?

Puzzlemix explains visibility from the clue's viewpoint: a taller building remains visible beyond a shorter one, while a shorter building behind a taller one is hidden. Read a top clue downward, a bottom clue upward, a left clue rightward and a right clue leftward.

Here is a separate original example. Reading 2, 1, 4, 3 from the left, you see the 2, then the 4. The 1 hides behind the 2; the 3 hides behind the 4. The count is 2, although the heights total 10. Adding is the wrong job for this clue.

Try this original 4x4 puzzle

Rows run top to bottom; columns run left to right. A question mark is an undecided cell. The top and bottom clue lists below are both written in column order, C1 through C4.

Top clues: 3, 2, 2, 1. Bottom clues: 2, 3, 1, 3.

Copy the grid and edge lists if you want to work independently. The solution begins in the next section. Use enlarged text, paper or notes as you prefer; there is no timing requirement and no claim about anyone's intelligence or health.

Step 1: Complete two rows

R1's left clue is 4. Seeing all four different heights requires them to rise in order from that side. Therefore R1 is 1, 2, 3, 4.

R3's right clue is also 4. This time the rising order runs from right to left, so the row written in normal column order is 4, 3, 2, 1.

Keep the viewing direction beside the deduction. Reversing a correct sequence is still an incorrect entry, however politely the numbers line up.

Step 2: Place the bottom of column 3

C3's bottom clue is 1. The nearest building must be the tallest: any shorter first building would leave the 4 visible somewhere beyond it, producing at least two visible buildings. Put 4 in R4C3.

C3 now reads 3, ?, 2, 4. Only height 1 is missing, so R2C3 is 1. Coordinates make the explanation easier to audit: R2C3 means row 2, column 3.

Step 3: Settle column 2

C2 currently reads 2, ?, 3, ?. Its missing heights are 1 and 4, and its top clue is 2.

If R2C2 were 1, the column would become 2, 1, 3, 4. From the top, 2, 3 and 4 would be visible: three buildings, contradicting the clue. Therefore R2C2 is 4, and R4C2 is 1.

This checks both remaining placements, rather than selecting whichever looks balanced. Our systematic solving guide explains how to keep a temporary assumption separate from an established fact.

Step 4: Use the bottom clue of column 4

C4 reads 4, ?, 1, ?, with heights 2 and 3 missing. Its bottom clue is 3.

If the bottom cell were 3, you would read 3, 1, 2, 4 upward. Only 3 and 4 would be visible, giving two. That fails. Put 2 in R4C4 and 3 in R2C4 instead. Looking upward now gives 2, 1, 3, 4, with three visible buildings.

R2 now needs 2 in its first cell. R4 needs 3 in its first cell. The grid is complete:

     C1 C2 C3 C4
R1:   1  2  3  4
R2:   2  4  1  3
R3:   4  3  2  1
R4:   3  1  4  2

How can you verify the answer?

Each row and column contains 1, 2, 3 and 4 once. Reading visibility in row or column order gives:

All edge clues match. We also enumerated the row arrangements independently of the walkthrough. The paired left/right clues allow 1, 6, 1 and 6 possible rows respectively: 1 × 6 × 1 × 6 = 36 candidate grids. Exactly one satisfies the column requirements and remaining clues.

Correctness and uniqueness answer different questions: whether this grid works, and whether another grid also works. Both were checked here. For a different kind of crossing-line exercise, try our worked nonogram; its filled-block rules belong to that puzzle, not this one.

Sources

• Conceptis: Classic Skyscrapers rules.
• Puzzlemix: Skyscraper rules and visibility.

The practice grid, separate line example, walkthrough and enumeration are original to this article.