What the Numbers Say Before You Start

All of this is available on an untouched board and none of it ever has to be worked out twice. It is the whole opening, and on an easy board it is most of the solve.

The 1 and the staircase

A 1 is the best thing a rim can carry. One rooftop in sight means the first building blocks the entire street, and the only building that can block an entire street is the tallest one there is — so the lot against that edge takes the top height and nothing else was ever possible there. Write it in and move on.

A number equal to the side is the same fact upside down. Every roof in sight means nothing is ever hidden, which can only happen if each building is taller than the one in front of it, which on a complete set of heights means they climb one at a time from where you are standing. The whole street is settled by one number, and on a small board that can be a fifth of the answer.

Those two are the only rim numbers that decide anything outright, which is why boards carrying one feel so much gentler than boards that do not. It is also why the generator, when it is aiming at an easy board, so often keeps them: they are the numbers whose removal costs the most.

Before — a 1 printed under the first column



3

	1	4

After — the tallest tower goes against that edge



3
	4
	1	4

One rooftop in sight from below means the nearest building hides the other three completely, and on a board of side four the only height that can hide a 3 is a 4. The lot at the bottom of the first column takes it, and nothing on the board had to be read to know that — only the number itself.

what the reasoning read what it decides what it wrote

The cap every other number puts on

Between those extremes every number still says something, and it is the same thing every time. To see k buildings from an edge you need k−1 of them to appear after whichever one you are looking at, each taller than the last. So the lot nearest the edge cannot be taller than n−k+1, the next cannot exceed n−k+2, and so on until the cap stops being a cap at all.

Worked on a five with a 3 against it: the nearest lot is at most 3, the one behind it at most 4, and after that no restriction. The tallest tower is therefore banned from the first two lots of that street, which sounds small and is not — it is two eliminations on an empty board, available before anybody has thought about anything, and it is what makes the second rung start producing answers.

The caps are worth taking in one sweep round all four sides rather than one number at a time. They never change, they never need revisiting, and a board with the caps written in looks very different from the same board without them. Beginners who find this puzzle impenetrable have almost always skipped this step and gone straight to staring at the middle.

Before — a 4 printed under the last column

4
1


				4

After — the lot against that edge is capped at 1

4
1

				1
				4

Four rooftops in sight on a board of side four means every building is taller than the one in front of it, so the street climbs one storey at a time from the bottom edge upwards. The cap on the nearest lot is therefore one, which on a board whose heights start at one settles it outright, and the rest of the staircase follows.

Why this rung is not a technique so much as a reading

Everything above reads printed numbers and nothing else. It looks at no candidate, depends on no previous move, and once it has been done it has been done — which is exactly what makes it the opening rather than a tool. The other five rungs all read the board as it currently stands, and all of them can have something new to say after every single move.

It also means an easy board here is defined by how far this gets you. If the rim, plus ordinary crossing out, fills the whole grid, the board is easy — not because it looks sparse or busy, but because nothing above the second rung was ever needed.

Where this sits on the ladder

Rung 1Reading the rim
A 1 is the tallest tower standing next to it, and every other number caps the cells behind it.
Rung 2Singles
A height that is settled is gone from the rest of its row and its column.
Rung 3Only one home
A height that fits just one cell of a row or column belongs in that cell.
Rung 4Blocked heights
Put a height in a cell and ask what the view from that edge could still be.
Rung 5Counting the line
Write out the orderings a row still allows and keep only what every one of them agrees on.
Rung 6Both ends at once
The numbers facing each other across one line are a single constraint, not two.
Rung 7Assumption
Write a height in, follow it until the grid breaks, and rub it out again.

Common questions

What is the strongest number a skyscrapers rim can carry?

A 1, which settles the lot beside it outright. A number equal to the side is nearly as good, since it lays the whole street out as a staircase in one move.

Does a 2 on the rim tell me anything at all?

Yes — the lot nearest that edge cannot hold the tallest tower, because a tallest tower there would hide everything and the view would be one. That single elimination often starts the whole board.

Do I have to redo this after every move?

No, and that is what makes it the cheapest rung. The rim never changes, so everything it says on its own is said once. Reading it against the heights still standing is a later and much more expensive move.