Painting on Spec, and Why We Refuse It

It finishes every board ever printed, which is precisely the problem with it. Here is how to do it properly if you meet one elsewhere, and why nothing here will make you.

What a trial looks like on a painted board

Pick an undecided cell and provisionally paint it out. Follow the consequences with every ordinary argument — its neighbours survive, their twins go, chase all of it, check the corners — until either the board finishes or something breaks: two painted cells touching, two identical survivors in one line, a group of numbers walled off from the rest. If it breaks, that cell cannot be painted, and keeping it is a sound permanent deduction.

The discipline is entirely in the bookkeeping. Everything derived under a trial is provisional and must come off again if the trial fails, so it has to be visibly different from what you knew before — a second colour on paper, or the undo button here, which is exactly what it is for. A solver who cannot cleanly unwind a trial will eventually leave a consequence of something false on the board, and that is far worse than being stuck.

Choose the cell carefully if you are going to do it at all. The best candidate sits next to several cells that are already decided, because the consequences then arrive within a move or two and the branch resolves quickly. Trying a cell in an untouched part of the board produces a long chain that neither confirms nor contradicts anything for ages.

Before — nothing ordinary applies
	1	2	3	4	5	6
1	2	6	1	2	5	1
2	6	6	2	5	1	4
3	4	2	5	1	1	5
4	5	4	5	2	2	4
5	3	5	1	2	6	1
6	2	1	3	6	1	2

After — one cell, decided by contradiction
	1	2	3	4	5	6
1	2	6	1	2	5	1
2	6	6	2	5	1	4
3	4	2	5	1	1	5
4	5	4	5	2	2	4
5	3	5	1	2	6	1
6	2	1	3	6	1	2

This board has exactly one answer and no reasoned route to it — the kind this site grows, grades and then discards. Twenty-one ordinary moves have been made and everything above has run out: no decision left to chase, no corner closing, no pocket and no cut anywhere. Supposing one cell painted breaks the board, so it survives. Sound, and thoroughly unsatisfying.

what it decides what it wrote

Why a guess cannot be a difficulty

Because it works on everything without exception. Any board with a single answer can be finished by painting something, following it and unwinding, repeatedly if necessary, so "needs a guess" is a statement about the solver’s patience rather than about the board. A scale whose top rung fits every puzzle is not measuring anything at all.

It is also what a generator falls back on when it has stopped checking. Scatter some painted cells, put numbers on them that repeat the survivors, verify only that the answer is unique, and you get boards that are unique and unreachable by reasoning — which is the defect behind almost every "impossible" review a hitori app collects. The board is not impossible. It has simply stopped being a puzzle, and the app cannot tell, because it never asked.

So the generator here asks. Every board is put through the technique solver as well as the counter, and any that needed a guess is discarded and another grown in its place. About one board in ten fails that test, which is a great deal of thrown-away work and is the single decision this section is built around. The position below is what one of those rejected boards looks like part-way through.

When a board from elsewhere leaves you no choice

Trial into the most surrounded cell you can find, keep the provisional marks visibly separate, and stop the instant you hit a contradiction rather than pressing on to see how bad it gets. One decided cell is a complete result — go straight back to the cheap arguments, because that decision will usually paint out a twin or keep a neighbour and set off a cascade.

The moment you catch yourself starting a second trial inside the first one, stop and put the board away. Nobody unwinds two layers of provisional painting reliably, and a board asking for it was made by something that never once checked whether a person could get through it.

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

Is trial and error allowed in hitori?

It is sound reasoning — a supposition that leads to a broken rule proves the opposite. But it works on every board, so a board requiring it has not been checked for human solvability rather than being especially hard.

Do any boards on this site need a guess?

None. Roughly one board in ten comes out of the generator needing one, and every one of those is thrown away and replaced rather than shipped under a harder label.

Why do some hitori apps produce boards that feel impossible?

Because they check only that the answer is unique. Unique and humanly reachable are different properties, and a generator testing the first without the second will ship boards only a trial can finish.