### Skyscrapers

Posted:

**Wed 16 Dec, 2020 8:15 pm**Ahead of Round 4 of Division D, I wanted to look at Skyscrapers puzzles.

GM Puzzles give the rules as follows:

Insert a digit from 1 to N into each cell in the N by N grid so that no digit repeats in any row or column. Also, each number in the grid represents the height of a building and the clues on the outside of the grid indicate how many buildings can be “seen” when looking from that direction. Taller buildings block the view of smaller buildings. For example, if a row contained the numbers 15342, then two buildings are seen from the left – 1 and 5 – and three buildings from the right – 2, 4, and 5 – with the other buildings blocked by taller buildings in front of them.

There are a number of things to think about with a skyscrapers puzzle, and I hope to highlight the key ones in this example.

In this 4x4 puzzle, clues of 4 are the biggest giveaway as there's only one way to see all four skyscrapers.

Clues of 1 always mean that the biggest number must sit right in front of it, as the biggest tower can never be hidden in a row.

The biggest towers can often be the easiest to think about. There are two places that the 4 could go on the top row. It can't sit right in front of the 2 clue, because then it would only see one skyscraper, so it must go in the red position.

There must be four 4s in the grid, and there's now only one place for the remaining 4.

Row two needs a 3 and a 1. If the 3 was in the pink position, then the 2 clue at the top would see three skyscrapers, regardless of the order of the remaining skyscrapers in that column, so that isn't right.

The 1 and the 3 must be this way round then.

For the 3 clue on the bottom to hold, the 1 and 2 in this column must be this way around.

Row 3 can be completed with the missing number and Row 4, Column 3 must be 3 otherwise there would be a repeat of 1, 2 or 4 in a row or column.

This kind of logic fills the rest of the grid.

GM Puzzles give the rules as follows:

Insert a digit from 1 to N into each cell in the N by N grid so that no digit repeats in any row or column. Also, each number in the grid represents the height of a building and the clues on the outside of the grid indicate how many buildings can be “seen” when looking from that direction. Taller buildings block the view of smaller buildings. For example, if a row contained the numbers 15342, then two buildings are seen from the left – 1 and 5 – and three buildings from the right – 2, 4, and 5 – with the other buildings blocked by taller buildings in front of them.

There are a number of things to think about with a skyscrapers puzzle, and I hope to highlight the key ones in this example.

In this 4x4 puzzle, clues of 4 are the biggest giveaway as there's only one way to see all four skyscrapers.

Clues of 1 always mean that the biggest number must sit right in front of it, as the biggest tower can never be hidden in a row.

The biggest towers can often be the easiest to think about. There are two places that the 4 could go on the top row. It can't sit right in front of the 2 clue, because then it would only see one skyscraper, so it must go in the red position.

There must be four 4s in the grid, and there's now only one place for the remaining 4.

Row two needs a 3 and a 1. If the 3 was in the pink position, then the 2 clue at the top would see three skyscrapers, regardless of the order of the remaining skyscrapers in that column, so that isn't right.

The 1 and the 3 must be this way round then.

For the 3 clue on the bottom to hold, the 1 and 2 in this column must be this way around.

Row 3 can be completed with the missing number and Row 4, Column 3 must be 3 otherwise there would be a repeat of 1, 2 or 4 in a row or column.

This kind of logic fills the rest of the grid.