Pockets You Have to Count

Between a corner you recognise and a board you have to trace lies the pocket: five or six numbers, sealed by a single cell, and nothing about it looks familiar.

Too big to see, small enough to follow

The argument is identical to the corner one — painting a cell that would seal a group of survivors away from the rest is impossible, so the cell is kept. What changes is the size of the group, and size is what makes it hard. Two or three cells in a corner is a shape you know. Six cells reaching round a bend behind a line of painted cells is not a shape at all, and the only way to be sure is to follow it.

The way to follow it is to pick the suspect cell, cover it with a finger, and walk outwards from one of the survivors beside it, stepping only onto cells that are not painted. If the walk comes back with fewer survivors than the board holds, you have found a pocket and the suspect cell is kept. It is slow, it is reliable, and it is the point at which most people start drawing on the paper.

What makes a cell suspect is worth learning, because trying every cell is unbearable. Look for a cell with painted cells on two opposite sides, or a cell that is the only gap in an otherwise unbroken line of painted ones. Those are necks, and a pocket always sits behind a neck.

Before — the top row reaches the rest of the board through one cell
	1	2	3	4	5
1	2	4	3	5	1
2	3	2	5	3	2
3	4	5	1	1	2
4	1	3	4	2	5
5	3	2	2	3	2

After — that cell is kept
	1	2	3	4	5
1	2	4	3	5	1
2	3	2	5	3	2
3	4	5	1	1	2
4	1	3	4	2	5
5	3	2	2	3	2

Painting the 2 in the top-left corner would leave the six numbers along the top of the board reachable only through cells that are already painted, cutting them off from the eleven cells below. That cannot be part of any answer, so the 2 survives — and finding it meant walking the region rather than recognising a shape.

what it decides what it wrote

The half of the board that is not the pocket

A useful trick is to walk the larger side rather than the smaller one. If you start from a survivor far away from the suspect cell and the walk reaches everything except a handful of cells huddled behind it, you have proved the same thing with less risk of miscounting — and you will usually notice the shortfall long before you finish, because the region you are tracing stops growing.

The other habit worth building is to stop as soon as you know. The question is whether the pocket exists, not how big it is. The moment your walk has failed to reach a survivor it could have reached, the deduction is complete and the suspect cell can be marked as kept.

And when a pocket is found, chase it. The cell you just kept paints out its own twins, keeps neighbours, and very frequently opens the pocket up entirely — pockets tend to be where the rest of the board has been waiting.

The bottleneck that makes a board hard

A hard board here is one where the solver, always taking the cheapest available move, was at some point forced into this — no decision left to chase, no corner about to be sealed, and a pocket somewhere that had to be traced rather than recognised. That is a statement about the board containing a genuine bottleneck rather than about its size.

It is also why size and difficulty are separate choices on this site. A 10×10 easy board is a hundred cells and no bottleneck at all; a 6×6 hard board is thirty-six cells and at least one. Most people find the second takes longer, and nearly all of them find it more interesting.

Where this sits on the ladder

Rung 1Printed patterns
Twins side by side, and a number sandwiched between two of its own, decide themselves.
Rung 2What a decision forces
A painted cell keeps its neighbours, a kept cell paints out its twins, and a cell with one way left keeps it.
Rung 3Corners and edges
Painting that would wall two or three numbers into a corner is painting you cannot do.
Rung 4Sealed pockets
The same argument on a pocket too big to recognise and small enough to count.
Rung 5Cutting the board
A cell whose painting would divide the surviving numbers into two halves.
Rung 6Assumption
Paint a cell, follow it until the board breaks, and scrub it out again.

Common questions

How do I know which cells are worth checking?

Look for necks: a cell with painted cells on opposite sides, or the only gap in a line of painted ones. A pocket always sits behind a neck, so those are the only cells worth tracing from.

Should I trace the pocket or the rest of the board?

Usually the rest. Walking outwards from a survivor far away and noticing that the region stops growing before it has reached everything is quicker and easier to get right than counting a small huddle.

What should I do once I have found one?

Chase the cell you just kept. It paints out its own twins and keeps its neighbours, and a pocket is very often where the rest of the board has been waiting.