Numbers You Would Wall In

The connectedness rule at its cheapest, and the first place a board gives way once the cascades have run dry. Corners go first because corners run out of room first.

Count the ways out before you paint

Everything that survives has to reach everything else that survives. A cell in the middle of the board has four ways out, an edge cell has three, and a corner cell has two — so a corner is where that requirement starts to bite long before anywhere else does. Painting the two cells that cover a corner seals it off, and a sealed corner cannot be part of any correct answer.

The move, then, is to look at a cell you are about to paint and ask what it would cut off. If the answer is two or three numbers stuck in a corner or a notch along an edge, with nothing but painted cells between them and the rest of the board, that painting is impossible — so the cell survives, and you mark it as kept.

It is worth its own rung because the shapes are finite. There are only so many ways to seal off a corner, and after a dozen boards you stop counting anything and simply see them. That is what a cheap technique means on this site: not that the reasoning is shallow, but that it costs you almost nothing to spot.

Before — the top-right corner has one way in and out
	1	2	3	4	5
1	4	1	5	2	3
2	1	5	2	1	5
3	3	2	3	5	5
4	5	5	3	4	1
5	5	3	5	1	3

After — the cell that would seal it is kept
	1	2	3	4	5
1	4	1	5	2	3
2	1	5	2	1	5
3	3	2	3	5	5
4	5	5	3	4	1
5	5	3	5	1	3

By this point the painted cells have reduced the top-right corner to three numbers reachable only through the 5 at row 1, column 3. Painting that 5 would wall those three off from the other seventeen, which no correct answer can do — so it survives, and it is marked as kept rather than left undecided.

what it decides what it wrote

Which corners, and the one that is not a corner

Sweep the four corners after every few decisions, then the edges. A corner cell that has been kept and has one painted neighbour is one move from being trapped, and the cell that would trap it is decided the moment you notice. Boards that will not move under the cascades give way here more often than anywhere else.

The version people miss is the notch: not a corner of the board but a corner of the surviving region, where two painted cells already meet at a diagonal and a third would close the gap between them. It is the same argument and it looks nothing like a corner, because it is somewhere in the middle of the grid.

Do not confuse this with counting a surviving cell’s neighbours, which is a cheaper move from the rung below — that one asks what a cell you have already kept still has left, and this one asks what a cell you have not yet painted would take away. They arrive at similar-looking conclusions from opposite directions, and only one of them is available when nothing nearby has been decided.

The threshold that medium marks

A medium board here is one the solver finishes with the printed patterns, the cascades and this corner argument, and nothing wider. It is a real threshold rather than a gradual one: an easy board never needs it at all, and a medium board contains at least one moment where nothing whatsoever happens until somebody asks what a painting would seal off.

The moment is usually recognisable once you know its shape. Every decision has been chased to its end, nothing repeats that has not already been dealt with, and there is a corner of the board with one painted cell beside it that you have stopped looking at.

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

Why do corners matter more than the middle of the board?

Because a corner cell has only two neighbours and an edge cell three, while a middle cell has four. Fewer ways out means running short of them sooner, so corners are where the connectedness rule bites first.

How is this different from counting a kept cell’s neighbours?

That asks what a cell you have already kept still has available, and it needs the cell to be decided. This asks what a cell you have not yet painted would take away, and it works on a corner nobody has touched.

Does the sealed-off piece have to be in a corner of the grid?

No. It is just as often a notch in the middle where two painted cells meet diagonally, which is the version people overlook precisely because it does not look like a corner.