Cutting a Line Into Pockets

The first argument that reads the middle of a line, and the first one that pays you back for every cross you bothered to write down.

Deal the numbers out among the pockets

Crossed cells are walls. Walk along a line and the walls cut it into pockets of consecutive undecided or filled cells, each with a length. The numbers then have to be dealt out among those pockets in order: the first number into the first pocket that can take it, the second into that pocket or a later one, and so on. No run may straddle a wall, because a run is consecutive by definition.

What this buys you is usually an elimination rather than a fill. A pocket of two cells cannot hold a five; if every number that could reach it is bigger than two, the whole pocket is blank and can be crossed off outright. A pocket that comes before the first number could possibly reach is blank for the same reason. Those crosses then shorten the line again, and the cheaper rungs get another turn.

It also fills cells. Once a pocket is the only one a particular number can be in, that number is effectively a line of its own inside the pocket — and the overlap arithmetic applies to the pocket exactly as it applied to the whole line. A four confined to a five-cell pocket gives you three cells.

Before — a cross cuts the third column in two
	2	2	2	21	21
2
12
2
21
2

After — the pocket that cannot hold the run is crossed off
	2	2	2	21	21
2
12
2
21
2

The third column reads 2 and a cell part-way down it has already been proved blank, which leaves two pockets. One of them is a single cell and cannot hold a run of two, so every cell in it is blank as well. Nothing about the filled cells in that column was used — only where it had been cut.

what the reasoning read what it decides what it wrote

Where the walls come from, and why so many are missing

Almost all of them come from lines you have finished. A row whose numbers are entirely accounted for has every remaining cell blank; crossing them is free and takes seconds, and each cross is a wall in a column that has nothing else going for it. The commonest reason a solver’s board stops moving is that this was skipped on four or five rows an hour ago.

The rest come from the argument on the page before this one, which crosses cells at the ends of lines, and from single cells proved blank because no run can reach them. All of them are cheap to make and expensive to be without.

On paper the habit worth building is to cross a whole line off the moment its last run goes in, rather than moving on to the next interesting thing. It feels like tidying. It is the setup for every deduction in the second half of the board.

What a medium board is promising you

A medium board here is one the solver finishes with the numbers, the ends of the lines, and this dealing out of numbers among pockets, and nothing wider. It is a real threshold rather than a gradual one: an easy board never needs it, and a medium board contains at least one moment where nothing whatever happens until somebody looks at where a line has been cut.

The moment is recognisable once you know its shape. Every long number has already given what it can, the ends of every line have been worked in from, and there is a row somewhere with a couple of crosses in the middle of it that nobody has thought about since they were written.

Where this sits on the ladder

Rung 1Overlap
A run longer than half the line has to cover the middle of it, wherever it sits.
Rung 2Working in from the ends
What the settled cells at the two ends of a line force, without looking between them.
Rung 3Gaps and splitting
The crosses cut the line into pockets, and each number has to fit into one of them.
Rung 4Glue
A filled block you can see at a glance belongs to a run, and that says how far it reaches.
Rung 5The whole line at once
One or two filled cells adrift in an open stretch, read with everything else on the line.
Rung 6Assumption
Fill a cell on spec, follow it until a line breaks, and rub it out again.

Common questions

What exactly is a pocket?

A stretch of consecutive cells that are not crossed off, bounded by crosses or by the ends of the line. A run must sit inside one pocket, because a run is consecutive and a cross cannot be part of it.

Why do I never seem to have enough crosses?

Because finished lines are rarely crossed off. When a line’s numbers are all placed, every remaining cell in it is blank, and each of those is a wall in a line running the other way.

Can a pocket be too big to be useful?

It can be too big to eliminate, but even then it helps: once a number is confined to a single pocket, the overlap arithmetic applies inside that pocket as though it were a line of its own.