The Cell That Would Halve the Board

The connectedness rule at full stretch. There is no shape to recognise here and no pocket to count — the two pieces are both large, and only a trace will find the neck.

A neck in the middle of a shape with no edges

By the time this argument is needed the board looks finished and is not. Every repeat has been dealt with, every decision has been chased, no corner is about to close, and the surviving region has grown into a sprawl with inlets and peninsulas. Somewhere in it there is a single undecided cell that everything on one side has to pass through, and painting it would leave the board as two separate territories.

What makes this the hardest published rung is that there is nothing local to look at. A corner announces itself and a pocket sits behind a visible neck; a cut that halves the board is a neck in the middle of a shape that has no reason to be tidy, with a dozen cells on one side and a dozen on the other. You cannot see it. You have to trace the region and watch it come apart.

The method is the same as for a pocket and the discipline is different. Pick the cell, cover it, and walk the survivors. What you are watching for is not a huddle left behind but the walk simply stopping — arriving at a wall of painted cells with half the board still unvisited on the other side of it.

Before — the surviving numbers meet at one undecided cell
	1	2	3	4	5	6
1	2	2	3	1	6	5
2	4	1	5	3	3	5
3	1	3	4	4	1	1
4	3	2	6	5	1	4
5	1	5	4	1	2	2
6	4	6	5	2	4	3

After — the cell holding the board together is kept
	1	2	3	4	5	6
1	2	2	3	1	6	5
2	4	1	5	3	3	5
3	1	3	4	4	1	1
4	3	2	6	5	1	4
5	1	5	4	1	2	2
6	4	6	5	2	4	3

The painted cells have grown into two diagonal chains reaching towards each other across the middle of the board, and the 2 at row 4, column 2 is the only gap left between them. Painting it would leave eleven cells above and fifteen below with no route between the two, so it is kept.

what it decides what it wrote

Finding the cell worth testing

Trying every undecided cell is not a technique, it is an afternoon. What narrows it down is looking at the painted cells rather than the surviving ones. Painted cells cannot touch, so they never form a solid wall — but two diagonal runs of them coming towards each other from opposite sides of the board leave exactly one gap where they nearly meet, and that gap is the cell to test.

Look, therefore, for the long diagonal chains. They are the characteristic shape of a hitori answer and they are what divides a board when they run far enough. If two of them are pointing at each other with one undecided cell between them, you have found your candidate without tracing anything.

And when the trace confirms it, the payoff is usually large. A cell that was holding the whole board together is a cell with a great deal depending on it, and keeping it tends to settle several others immediately through its own twins and neighbours.

Why nothing published stands above this

Above it there is only the guess — paint something, follow it until the board breaks, scrub 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 cell that nothing else will decide, and the reason it is decided is that painting it would divide the survivors in two. Boards needing anything beyond that are discarded during generation rather than sold as a harder level.

On a small board it is genuinely hard to produce. Twenty-five cells do not leave much room for two halves, so a 5×5 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

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 find the cell that would cut the board?

Look at the painted cells rather than the survivors. They form long diagonal chains, and where two chains point at each other with a single undecided cell between them is the gap worth testing.

How is this different from finding a sealed pocket?

A pocket leaves a small huddle behind a visible neck. A cut leaves two large territories, the neck is somewhere in the middle of an untidy shape, and nothing about it is visible until you trace.

Why do small boards rarely come out at expert?

Because twenty-five cells do not leave room for two substantial halves. A 5×5 asked for at expert often measures as hard instead, and it ships under the label it measured.