Testing a Tower Against a View

The first move that reads a printed number against the board rather than against the whole range of heights, and the one that carries the middle of every board here.

Two sweeps, and what they bracket

Suppose a height into a lot on a street that carries a number, and then ask two questions about the street with that height standing in it. How many rooftops could you see at most? How many at the very fewest? Those two answers bracket the view, and if the printed number is outside the bracket, the height cannot stand there.

The most you could see is worked out by keeping the skyline as low as possible: walk the street from the edge, take the shortest height still available at each lot, and count every lot whose tallest remaining height clears the roofline behind you. The fewest is the mirror image — keep the skyline as high as possible and count only the lots that clear it even so. Neither sweep is exact and neither needs to be, because a number outside the bracket is impossible whatever the exact answer turns out to be.

In practice nobody phrases it like that. What people say is "a 5 there would block the whole street and it says three", or "a 1 there hides behind anything, so the three I still need have to come out of two lots". Both of those are this argument, and both are one sweep along a street rather than a piece of paper.

Before — a 3 under the first column, and a lot that could be 3 or 4



4	1	2	3	4
	2	4	1	3
	3		3

After — the taller of the two is impossible


	3
4	1	2	3	4
	2	4	1	3
	3		3

Looking up the first column from the bottom the street reads 2, then 1, then the lot in question, then the top lot. A 4 standing there would put the tallest tower third in line, hiding the top lot completely, and only two roofs would be in sight. The number says three, so the 4 goes and the 3 stays.

what the reasoning read what it decides what it wrote

Where to point it

At the tall heights first, and against the edges nearest them. A tall tower placed close to an edge is the most destructive thing that can happen to a view — it hides everything behind it — so it is the placement most likely to contradict a number, and a number of 2 or 3 on a street of five or six is the number most likely to catch it.

Then at the low heights near an edge with a large number. A 1 standing next to an edge that wants four rooftops has spent one of the four on a single-storey building and left the other three to be found among fewer lots than before, which very often turns out to be impossible.

What is rarely worth testing is a middling height in the middle of a street with a middling number against it. The bracket there is wide, the sweep tells you nothing, and time spent on it is time not spent on the two cases above. This is the rung where knowing where to look is most of the skill.

Before — a 3 facing the second row from the right

4		2
3	4	3
2
1
4

After — the nearest lot cannot be the taller of its two heights

4			2
3	4	1	3
2
1
4

Read the second row from the right and it runs: this lot, then a lot holding a 1 or a 2, then a 4, then a 3. A 2 in the nearest lot would leave the following lot holding a 1, hidden behind it, so only the 2 and the 4 would be in sight and the view would be two. The number says three, which forces the 1.

The exact line between easy and medium

A medium board here is one where the solver, always taking the cheapest available move, was at some point forced into this one. No printed number had anything left to say on its own, no lot was down to a single height, and no height had one home left in its line — the board did not move until a tower was tried somewhere and rejected by the view it would have produced.

It is a real threshold rather than a gradual one. An easy board never needs it even once; a medium board contains at least one moment where nothing happens at all until somebody imagines a building standing somewhere and works out what the street would then look like.

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

Is this the same as guessing?

No. A guess keeps what does not break; this keeps only what a contradiction rules out and writes nothing on the board that was not proved. Nothing provisional ever survives the move.

Which height should I test first in a skyscrapers puzzle?

The tallest one still available in a lot near an edge with a small number against it. A tall tower close to the edge hides the most and therefore contradicts the most.

Why is this easier than writing the street out?

Because it is two sweeps along a line rather than a list of arrangements. It sometimes misses what the full list would find, which is exactly why the full list is the rung above it.