Writing the Street Out

Two rungs and one piece of paper. Everything cheaper reads a lot, a height or a sweep; these two read the arrangement of a whole street, which is what the numbers are actually about.

Writing the orderings out, and keeping the list short

Take a street with a number at one end and write out every arrangement of heights that still fits: the candidates left in each lot, in every order they could come, keeping only the orders that produce the printed view. Then read down the list. Any lot holding the same height in every arrangement is settled. Any height appearing in no arrangement for a lot is gone from that lot, even though nothing has been settled anywhere.

That second half is the usual result and it is the more useful one. A street of five with three heights still floating routinely has four or five surviving arrangements, they rarely agree on a whole lot, and they very often agree that some particular height is nowhere — which is the elimination that lets a cheaper rung finish the job two moves later.

The list is only worth writing when it is short, and what shortens it is everything below this rung. A street with every lot wide open has a hundred and twenty orderings on a side of five and is not worth the paper; the same street with two lots settled has six. Choosing the shortest list on the board rather than the most annoying street is most of the skill here.

Before — a 3 facing the third row from the right
			1
			4
		4	3
4				3
	4
	1		3

After — one height leaves the middle of that row
			1
			4
		4	3
4				3
	4
	1		3

The third row holds a 4 at its far end, so that tower is always the last thing in view and the other two roofs must come from the three lots nearer the edge. Writing those three out, the orders that show exactly two roofs never put a 2 in the lot second from the right — so the 2 goes, and nothing else about the row has been decided.

what the reasoning read what it decides what it wrote

Both ends at once

A street with a number at each end is not two facts. It is one: the arrangements that show three roofs from the left AND two from the right, which is a far shorter list than either number produces alone. Reading them separately and then combining what each one gave you is strictly weaker, because the constraint is on the ordering rather than on any individual lot, and an ordering can satisfy each number in isolation without satisfying both.

This is the hardest thing any board here is allowed to need and it is what an expert board is made of. In practice it fires exactly where nothing else can: a street where each number alone still permits everything, and the pair of them permits three arrangements that happen to agree about one lot.

The pair also carries a free fact worth knowing before any list is written. Two numbers facing each other along a street can never total more than one plus the side, because the tallest tower is in sight from whichever end it is nearer and is counted only once between them. A 4 and a 4 on a side of five is not a hard board, it is an impossible one, and it would never be printed.

Before — two lots of the last column still holding a 1 or a 2
			1
			4
		4	3
4				3
	4
	1		3

After — the same list settles both of them
			1
			4
		4	3
4			1	3
	4		2
	1		3

The list for that row is now short enough that every surviving arrangement agrees about two lots at once, which is the pleasant case and the reason the arrangements are worth writing down rather than reasoned about one lot at a time. Two heights are settled by one reading, and the row that was stuck a moment ago is nearly finished.

And there the ladder stops

Above this there is only trial — write a height in, follow it until the board breaks, rub it out — and that will finish any board at all, which is exactly why it cannot be a difficulty level. A scale whose top rung fits everything measures nothing.

So hard here means a street had to be written out, and expert means a street had to be written out against both of its numbers together. Boards needing anything beyond that are stopped during generation rather than sold as a harder level.

Before — a 3 on the left of the top row and a 2 on the right

3						2
		5
5	1	2	3	4	5
						2
	5					3
		2			2

After — the fourth lot loses two of its heights

3						2
		5
5	1	2	3	4	5
						2
	5					3
		2			2

Neither number has anything left to say on its own here: read from the left the top row still allows the fourth lot almost anything, and read from the right it does too. The arrangements that show three roofs one way and two the other are far fewer, and every single one of them puts either the shortest tower or the tallest in that lot.

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

How do I know which street to write out?

The one with the shortest list, which usually means the most settled lots and a number near the extremes of what the side allows. A wide-open street on a large board is not worth the paper.

Do I have to write every ordering down?

Only the ones that fit. Prune as you go: the count of roofs in view can only rise, so an opening that has already exceeded the number is dead, and once the tallest tower is placed nothing behind it can ever be added.

Why read both ends of a street together?

Because the numbers constrain the ordering rather than any lot, and an ordering can satisfy each of them separately without satisfying both. Combining them afterwards throws that away.