Skyscraper-style puzzles and step-by-step deduction

A row of four numbers can look simple until someone asks how many buildings you could see from one end. That question is the heart of a skyscraper-style puzzle. You place building heights in a square grid, then use clues around its edges to work out where each height belongs. The fun comes from making one certain move and seeing what it tells you about another row.

On a 4×4 board, the heights are 1, 2, 3 and 4. Every row uses each height once. Every column does too. An outside clue tells you how many buildings are visible when you look along that row or column. A tall building hides any shorter building behind it (ukpuzzles.org guide; Puzzle Skyscrapers rules).

Count what you can see

Imagine the row 2, 4, 1, 3. Looking from the left, you see the 2 first. You also see the 4, because it is taller. The 1 and 3 sit behind the 4, so the left clue is 2. Looking from the right, you see the 3 and then the 4. The right clue is also 2. The two ends of a row can give different clues; this row happens to give 2 at both.

Two edge clues are especially useful. A clue of 1 means the tallest building must stand nearest that edge. Nothing is taller, so the 4 hides everything behind it. A clue of 4 means every building must be visible. The heights must rise in order: 1, 2, 3, 4, as you look inward (ukpuzzles.org guide).

A clue of 2 or 3 takes more work. It usually rules out several orders without giving you the whole row. Check both ends when you have both clues. Then check the crossing columns. That is where a small fact becomes a useful deduction.

Work through a 4×4 grid

Here is a teaching example. It is separate from any board in freegama. A dash is an empty square. The top clue for column 1 is 4. The left clue for row 1 is 4. Row 4 also has a left clue of 1. A few heights are already given.

Row	Column 1	Column 2	Column 3	Column 4
1	–	–	–	–
2	–	3	4	–
3	–	4	–	–
4	–	–	–	3

First, use the top clue. To see four buildings down column 1, they must rise 1, 2, 3, 4. Put those heights into rows 1 through 4 of column 1. The left clue of 1 on row 4 now checks out: its first building is 4, which hides everything behind it.

Next, use row 1's left clue. That row must also rise 1, 2, 3, 4. You now know its four cells. Notice that the two clues agree at row 1, column 1: both require a 1 there.

Now fill row 2. It reads 2, 3, 4, –. A row needs each height once, so the last cell is 1. This step needs no sightline arithmetic. It follows from the no-repeat rule.

Look at column 4. It now contains 4, 1, –, 3. Its missing height is 2, so row 3, column 4 is 2. Row 3 reads 3, 4, –, 2. Its missing height is 1.

Finish row 4. It reads 4, –, –, 3. Column 2 already has 2, 3 and 4, so row 4, column 2 must be 1. The last square in row 4 is 2. The finished rows are:

Row	Heights from left to right
1	1, 2, 3, 4
2	2, 3, 4, 1
3	3, 4, 1, 2
4	4, 1, 2, 3

Check the result from every shown edge. Looking down column 1, you see all four heights. Looking across row 1, you see all four. Looking across row 4 from the left, you see only its 4. The givens and the no-repeat rules also hold. A complete puzzle asks you to satisfy all its shown clues together, not just the clue that gave you your first move.

What are you practising?

You are keeping several rules in view, testing a possible height against them, and crossing out choices that cannot fit. The precise skill in this puzzle is deduction under these grid rules. You may become better at spotting a useful clue and checking a row without recounting it from scratch. That is a fair description of progress in this kind of game.

It would go further than the evidence to say that a tower puzzle raises your general reasoning ability. A scientific consensus on brain training says that people often improve on practised tasks, while evidence for broad benefits in everyday life is much weaker (Stanford Center on Longevity and Max Planck Institute, 2014). Enjoy the logic for what it asks you to do here.

Get faster without guessing

Start each board by scanning for 1s and clues equal to the grid size. Put in any certain heights, then check the crossing rows and columns for a missing number. When a cell still has two possible heights, keep both in mind or jot both down. Do not turn a possibility into an answer until another clue rules one out.

After a move, ask, “What changed?” Filling a 4 might finish an edge clue. Filling the last number in a column might settle a row. That question is more useful than scanning every square again. Speed can grow when you recognise the same kinds of deductions and check them carefully. Rushing past a clue can cost more time than taking a moment to count the visible buildings.

In Tower Sights, you work out the heights of two marked towers, A and B. Some heights and edge clues are shown; you do not need to fill every blank on screen. Try identifying the certain heights first, then use their rows and columns to reach A and B. The aim is to explain each answer to yourself before entering it.