When a Height Has Nowhere Left to Stand

Two rungs, one habit, and nobody learns either without the other. Between them they finish every easy board on this site and they carry the middle of every other one.

A settled height clears its street and its avenue

The moment a lot is known, that height is unavailable everywhere else along its row and everywhere else down its column. This is the bookkeeping the whole puzzle runs on, and it is the rung that produces almost every answer on a gentle board — not because it is clever but because it cascades. Cross a height out of four lots and one of them is very often left holding a single possibility, which is then a settled lot of its own.

It also covers the case beginners meet first without ever calling it a technique. A street with one empty lot takes the height the street is missing; four written heights on a side of five leave exactly one candidate, and no view is consulted at any point. That is this rule applied four times over and there is no reason to give it a name of its own.

On paper the discipline is to sweep rather than to hunt. Run your eye along every row and then down every column after each few moves, rather than checking the line you happen to be looking at. A lot with one height left sitting in a column you have not glanced at for ten moves is the single most common thing left undone on a stalled easy board.

Before — a 4 already standing in the third row

1	4

4	1	2	3	4
					3

After — the 4 leaves the lot above it

1	4

4	1	2	3	4
					3

The staircase written across the third row by the 4 on its left edge put a tower of four in the last column. Every other lot in that column therefore loses the 4, and the one directly above it — which until this moment could still have been anything — is down to three candidates without a single view being consulted.

what the reasoning read what it decides what it wrote

A height with one lot left

The other way round, and the harder of the two to see. Take a street and pick a height that is not standing in it yet. Look at every unsettled lot along that street and ask whether the height is still possible there. If exactly one lot survives the question, that is where it goes — and that lot may well still have three or four other candidates of its own, which is precisely why nobody notices.

That is what makes this move worth hunting deliberately rather than waiting for. A lot down to one candidate announces itself as you scan; a height down to one home announces nothing at all, and finding it means asking the question of each missing height in turn rather than looking at the lots. Five heights and five rows is twenty-five small questions on a five, most of them answered instantly.

The payoff is that it reaches places nothing cheaper does. A lot with four candidates is invisible to the rung below this one, and this argument can settle it outright — which on a medium board is very often the move that restarts everything after a stall.

Before — the tallest tower has one lot left in the top row

3

	4	3	2	1	4
				4	1

After — it is written there

3			4

	4	3	2	1	4
				4	1

Nothing in the top row is settled and three of its four lots still hold two or three candidates each, so no lot announces itself. But the 4 has been crossed out of all of them except the third, and a row has to contain a 4 somewhere — so that is where it stands, regardless of what else that lot could also have been.

Where the line between easy and medium falls

An easy board here is one the solver finishes using nothing above these two, and in practice most of its moves are the first of them. That is a definition rather than an approximation of one: the generator stopped rubbing numbers out the moment a removal would have needed anything harder.

Which also means an easy board is guaranteed finishable by a reader who knows only this page and the one before it. If yours has stalled, the overwhelmingly likely cause is a street with one gap in it that nobody has looked at, rather than anything requiring a technique you have not met.

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 a hidden single in a skyscrapers puzzle?

A height that has only one remaining lot in its row or column. The lot may still carry several other candidates, which is why the move is so easy to walk straight past.

Do I really need to write candidates into the lots?

For anything above an easy board, yes. Both halves of this technique are readings of crossing-out you have already done, and neither is visible if the crossing-out only happened in your head.

Which half of this is worth doing first?

Sweep for a lot with one height left, since it catches your eye on the way past. Go hunting for a height with one home only once the sweeping has stopped paying.