Edge clues: 1s, staircases and bounds
A 1 on the edge puts the tallest building right beside it. A clue equal to the size is a staircase. Every other clue caps how tall the nearest buildings can be.
How it works
A visibility clue counts the buildings you can see along its row or column. The tallest building is always visible, because nothing can hide it. Everything before it that is a new high also shows.
So a 1 means the first building is already the tallest: in a 5×5 it is a 5. A clue equal to the size means every building shows, so each is taller than the one before, and the line reads 1, 2, 3 and up.
Any other clue c works the same way, only weaker. The first building must leave room for c − 1 taller ones behind it. In an n×n grid it is at most n − c + 1, the next at most one more, and so on.
The bound, as a rule
With a clue c on an n×n grid, the cell k steps from the edge (counting the first as 1) holds at most n − c + k. A 2 on a 5×5 caps the first cell at 4, which just says the 5 is not first.
Why: if the first building were taller than that, there would not be enough taller heights left to make c − 1 more buildings visible. That is proof by contradiction in one line.
Where the tallest can go
The bound always rules the tallest building out of the cells nearest any clue bigger than 1. Mark those, and a row or column often has one place left for its tallest building.
Worth memorizing
A 1: the tallest building is at the edge.
It hides everything behind it.
A clue equal to the size: 1, 2, 3 and up.
Every building has to show.
A clue c caps the first cell at n − c + 1.
One more for each step in.
Practice
0 of 4 rightRow 1 shows 1 from the left. What height is in the ringed cell?
Column 4 shows 4 from the bottom. What is the tallest the ringed cell can be?
Row 5 shows 3 from the right. What is the tallest the ringed cell can be?
Row 3 shows 5 from the left. What height is in the ringed cell?