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Puzzle Quarry

One Cell Adrift, and the Line Around It

There is nothing local to look at here. The line gives way only when its crosses, its blocks and its stray cells are all held in mind together.

List the arrangements and keep what they agree on

The method is the definition of the puzzle taken literally. Pick one line, write down every way its numbers could legally be placed given everything already marked on it, and look at what all of those ways have in common. Any cell filled in all of them is filled; any cell blank in all of them is blank. Nothing is left over — this is the strongest anything can ever say about a single line.

Doing it by hand is less painful than it sounds because most arrangements die immediately. Start the first number at the first possible place, check it against the marks, and move it along; the moment a placement covers a crossed cell or leaves a filled cell uncovered, it is gone. On a line with two or three numbers there are rarely more than a handful of survivors.

What makes this expensive is not the arithmetic but the attention. Every other rung on this ladder lets you look at part of a line and reach a conclusion. This one does not — a single cell in the wrong column of your working and the answer is wrong rather than merely absent.

Before — one filled cell adrift in the second row
21122111
11
2
2
21
11
After — two more cells in that row are settled
21122111
11
2
2
21
11
The second row reads 2 and holds one filled cell with open cells on both sides of it. That cell cannot be glued to anything and its length rules nothing out, but writing down every placement of the two that still covers it leaves only arrangements agreeing about two further cells. Every mark on the row was needed to get there.

what the reasoning read what it decides what it wrote

The stray cell is the thing that forces it

What separates this rung from the one below is a filled cell, or a pair of them, sitting in the middle of an open stretch with nothing beside it. It cannot be glued, because there is no wall. Its length rules nothing out, because it is one cell. On its own it says almost nothing at all.

Combined with everything else on the line it can say a great deal, and that combination is precisely what has to be held in mind at once. The stray cell rules out the arrangements that do not cover it; the crosses rule out others; the blocks rule out more; and what survives all three filters may agree on cells that none of the three would have settled alone.

It follows that the useful reflex when a board is stuck is not to look for something clever but to find the line with the fewest survivors and enumerate them. The line with the least slack is usually the one, and the count of arrangements it still allows is often two or three.

The last rung anything here is allowed to need

Above it there is only the guess — fill something, follow it until a line 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 expert here means precisely this argument and no more: somewhere on the board there is a line that nothing cheaper will move, and the reason it moves at all is that every arrangement it still allows was written out and compared. Boards needing anything beyond that are discarded during generation rather than sold as a harder level, and drawings needing it were redrawn.

On a small board it is genuinely hard to produce, because a five-cell line rarely has enough room for a cell to be adrift in. A five asked for at expert quite often comes back labelled hard — that is the label the solver measured, and it ships under that rather than under the one requested.

Where this sits on the ladder

Common questions

How do I work a whole line out without making a mistake?

Place the first number at its first legal position, check it against every mark on the line, then slide it along. Most placements die at once, and the survivors are usually few enough to compare by eye.

Which line should I pick when the board is stuck?

The one with the least slack — its length minus its numbers and their compulsory gaps. Least slack means fewest arrangements, which means the shortest list to write out and compare.

Why is a single filled cell so much harder to use than a block?

Because its length rules nothing out and it has no wall beside it. A block of three can only belong to a run of three or more; a single cell could belong to any of them.

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