Trial, and the Line We Will Not Cross

It finishes every board ever made, which is the problem with it. Here is how to do it properly elsewhere, and why nothing here will ever make you.

How a trial is actually run

Pick an unsettled lot and provisionally stand one of its remaining towers in it. Follow the consequences with every ordinary argument — the rim, the crossing out, the singles, the blocked heights, the written-out streets — until either the board finishes or something breaks: a height twice in one street, a lot with nothing left in it, a printed number that no arrangement can now produce. If it breaks, that height was wrong, and removing it is a sound permanent result.

All the difficulty is in keeping the books. Every height you write under a supposition is provisional, and if the supposition fails then all of it has to come off again — so it has to look different from what you actually knew, which on paper means a second pencil and here means the undo button. A solver who cannot unwind a trial cleanly ends up with a consequence of something false left standing on the board, and being quietly wrong is a much worse place to be than being stuck.

Choose the lot carefully if you are going to do it at all. The best candidate is a lot with two heights left on a street that already carries a number, because the consequences arrive within a move or two and the branch resolves quickly. Trying a lot with five candidates in a quiet corner produces a long chain that neither confirms nor contradicts anything for ages.

Before — nothing ordinary applies anywhere
	3
3			4
		4			2
				4
	4				3
				2

After — one height, removed by contradiction
	3
3	2		4
		4			2
				4
	4				3
				2

This board has exactly one answer and no reasoned route to it — the kind this site carves, measures and then throws away. Eighteen ordinary moves have been made and nothing above is available: no printed number has anything left to say, no lot and no height has run out of room, and no street can be written out to any effect. Supposing the shorter of the two towers in the top-left corner breaks the board, so it is the taller one. Sound, and thoroughly unsatisfying.

what it decides what it wrote

Why this cannot be a difficulty at all

Because there is no board it fails on. Anything with a single answer falls to writing, following and unwinding, as many times over as you have patience for — so the sentence "this one needs a trial" is a fact about the solver rather than about the puzzle. A top rung that fits every board tells you nothing about any of them.

It is also exactly what a generator falls back on when it stops checking. Rub numbers off a rim at random, verify only that the answer is unique, and out come boards that are unique and unreachable by reasoning — which is the defect behind almost every one-star review a skyline puzzle app collects. The board is not impossible. It has simply stopped being a puzzle, and the app cannot tell, because it never asked.

So the generator here asks. Numbers are rubbed out one at a time and each removal is kept only while the technique solver can still follow the board without trial, which means the question is answered a few hundred times per board rather than once at the end. The position below comes from a board carved deliberately past that point: eighteen ordinary moves have been made, everything ordinary has run out, and only a trial will move it.

What to do if a book hands you one

Try into the most constrained lot you can find, keep the provisional heights visibly separate, and stop the instant you hit a contradiction rather than pressing on to see how bad it gets. One eliminated height is a complete result — go straight back to the cheap arguments, because that elimination has usually shortened a street’s list and settled a lot or two.

And the moment you notice a second supposition stacked on top of the first, close the book. Two layers of provisional working is where mistakes are made, and a board that demands it came out of a generator that never once asked whether a person could get through it.

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 trial and error allowed in a skyscrapers puzzle?

It is sound reasoning — a supposition leading to a broken rule proves the opposite. But it works on every board, so a board requiring it has not been checked for human solvability rather than being especially hard.

Does any board here need a trial?

No. Every removal from the rim is tested against the technique solver and the carve stops at the last one the solver could still follow, so a board needing trial is never reached in the first place.

Why do some skyline puzzle apps feel impossible?

Because they check only that the answer is unique. Unique and humanly reachable are different properties, and a generator testing the first without the second will happily ship boards only trial can finish.