Chasing a Decision

The least clever thing on this ladder and the one that decides the most cells. It is also the one people skip, which is why their boards stop moving.

Three consequences, one habit

You have just decided a cell. Write down what that forces, and there are only three things it can be. If you painted the cell, all four of its neighbours survive, because painted cells may not touch. If you kept the cell, every other copy of its number along its row and its column is painted out. And if a cell you have kept now has exactly one neighbour that is not painted, that neighbour survives too, because a survivor with no way out is cut off from everything.

They are three different rules pointing in three different directions and they are one habit, which is why they share a rung here. The question you are asking is always the same — I have decided this cell, so what does that settle? — and nobody who asks it asks only one of the three.

What makes this worth a rung rather than being called bookkeeping is the cascade. Painting a cell keeps four neighbours; each of those paints out its twins somewhere else on the board; each of those keeps four more. On an easy board that chain reaction is the entire solve, and every board at every level opens with a long run of it.

Before — the 1 in the middle of the board has been kept
	1	2	3	4	5
1	1	2	1	5	3
2	2	1	1	5	5
3	5	1	3	4	2
4	4	2	5	1	1
5	1	5	2	1	5

After — the other 1 in its column is painted out
	1	2	3	4	5
1	1	2	1	5	3
2	2	1	1	5	5
3	5	1	3	4	2
4	4	2	5	1	1
5	1	5	2	1	5

The cell at row 3, column 2 was settled as a survivor a moment earlier, and it holds a 1. No other 1 in its column may survive alongside it, so the 1 two rows above is painted out — which in turn will keep all four of that cell’s neighbours, and the chain carries on from there.

what the reasoning read what it decides what it wrote

Chase from where you last were

Work outwards from the cell you decided most recently rather than starting at the top left each time. That cell is the only place new information has entered the board, so it is the only place a new consequence can come from, and re-walking the whole grid to discover that is the difference between a five-minute board and a twenty-minute one.

When the run finally stops, go back and check the cells you decided three or four moves ago rather than reaching for something harder. A cell whose consequences were followed when half its neighbours were still undecided very often has more to say now that they are not. This is the single most common thing a stuck solver has left undone.

On paper, mark both kinds of decision differently and mark them as you go. A cell you have painted is obvious; a cell you have decided to keep looks exactly like a cell you have not thought about, and if it is not marked you will not chase it — which loses you the entire second consequence in the list above.

Why this plus the patterns is exactly what easy means

An easy board on this site is one the solver finishes using nothing but the printed patterns and this chasing. That is the definition rather than an approximation of it: the solver always takes the cheapest available step, and if it never needed anything above the second rung, the board is easy and is labelled so.

It also means an easy board is guaranteed to be finishable by a reader who knows only this page and the one before it. If yours has stalled, some decision has not been chased — and it is far more likely to be a cell you meant to keep than a cell you painted, because a painted cell is impossible to forget and a kept one is nearly invisible.

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

What does painting a cell tell me?

That all four of its neighbours survive, because two painted cells may never touch. It is four cells decided from one decision and it is the fastest move on the board.

Why should I bother marking cells I mean to keep?

Because a kept cell paints out every twin of its number along both of its lines. An unmarked one looks identical to a cell nobody has considered, so its consequences never get chased.

Where should I look after making a decision?

At that cell and what touches it. New information only enters the board where you just decided something, so anywhere else is a wasted sweep.