The Rules of Hitori
Three rules. The first says what to paint out, the second and third say what you cannot, and the third is the one that makes this a puzzle.
Rule one: no repeats among the survivors
When you have finished, look along any row. The numbers still standing must all be different from one another. The same goes for every column. Numbers you have painted out do not take part in this at all — a row can end with four survivors and six painted cells, and only the four have to differ.
This is the rule that tells you what needs removing, and it is the only one of the three that does. Every painted cell on a correct board is there because leaving it would have put two of the same number in one line. If you ever find yourself painting a cell that repeats nothing, you have made a mistake, because there was no reason to remove it.
It is worth noticing what the rule does not say: it does not say each number appears once. A 7×7 board might hold four 3s in one row, and the answer will keep exactly one of them and paint out the other three — or, if some other line forces it, paint out all four and keep none. Zero survivors of a value in a line is perfectly correct.
| 4 | 5 | 2 | 3 | 4 |
| 4 | 3 | 4 | 2 | 3 |
| 5 | 4 | 4 | 1 | 3 |
| 5 | 1 | 5 | 2 | 2 |
| 2 | 4 | 3 | 5 | 4 |
| 4 | 5 | 2 | 3 | 4 |
| 4 | 3 | 4 | 2 | 3 |
| 5 | 4 | 4 | 1 | 3 |
| 5 | 1 | 5 | 2 | 2 |
| 2 | 4 | 3 | 5 | 4 |
Rule two: painted cells may not touch
No two painted cells may share an edge. Diagonal contact is fine and is everywhere on a finished board, but two painted cells side by side or one directly above the other is illegal, however well the numbers work out.
The consequence you use constantly is the reverse reading: every neighbour of a cell you have painted survives. That is four cells decided at a stroke from a single decision, and it is the engine of nearly every chain of reasoning on the board — a survivor forces its twins to be painted, each of those keeps four more neighbours, and so on across the grid.
It also puts a ceiling on how much of the board can be removed. Painted cells form a scattered pattern rather than a blob, so on any board a clear majority of the numbers survive. A reader who has painted out half a row has almost certainly gone wrong.
| 4 | 5 | 2 | 3 | 4 |
| 4 | 3 | 4 | 2 | 3 |
| 5 | 4 | 4 | 1 | 3 |
| 5 | 1 | 5 | 2 | 2 |
| 2 | 4 | 3 | 5 | 4 |
Rule three: everything left has to stay joined
The surviving numbers must form one connected region. From any one of them you must be able to reach any other by stepping between cells that share an edge, never crossing a painted cell. Diagonal steps do not count here either.
Unlike the first two, this rule is not about any particular cell and cannot be checked by looking at one. A board can satisfy every repeat and every adjacency and still be wrong, because the painted cells happen to have fenced three numbers into a corner. Nothing local will tell you that has happened; you have to look at the shape of what is left.
Almost all the genuinely hard reasoning in hitori comes out of this rule applied backwards. Rather than checking a finished board, you ask of a cell you are thinking of painting whether painting it would seal anything off. If it would, it cannot be painted, so it survives — and that is a deduction available long before the board is anywhere near done.
| 4 | 5 | 2 | 3 | 4 |
| 4 | 3 | 4 | 2 | 3 |
| 5 | 4 | 4 | 1 | 3 |
| 5 | 1 | 5 | 2 | 2 |
| 2 | 4 | 3 | 5 | 4 |
| 4 | 5 | 2 | 3 | 4 |
| 4 | 3 | 4 | 2 | 3 |
| 5 | 4 | 4 | 1 | 3 |
| 5 | 1 | 5 | 2 | 2 |
| 2 | 4 | 3 | 5 | 4 |
Four things the rules get assumed to say
They do not say every repeated number must be painted out. Two 5s in a row means at least one goes; if there are three, at least two go. Which ones is the puzzle, and sometimes all of them go.
They do not say the surviving numbers form a rectangle, a border or any other tidy shape. A finished region is usually a sprawling blob with inlets, and the narrow necks in it are exactly where the third rule bites.
They say nothing about diagonals, in either direction. Painted cells may touch diagonally and surviving cells may not connect diagonally, which catches people out because the two rules feel like they ought to agree.
And they do not permit guessing. Every board published here was checked to be finishable by argument alone, so a board that will not move is one where something has been missed rather than one that wants a coin flipped. That is a promise about the generator, and it is why the generator throws so much of its work away.
Common questions
Can a whole row be painted out in hitori?
No, and not because of rule one — because of rule two. Painted cells cannot touch, so no line can have two adjacent painted cells, let alone all of them.
Do surviving numbers connect diagonally?
No. Connection means sharing an edge, so a survivor that touches the rest of the region only at a corner is sealed off and the board is wrong.
Is it possible for no copies of a number to survive in a row?
Yes, and it happens often. Rule one only forbids two survivors of the same value in a line; keeping none of them is entirely legal.
More Hitori pages
- Hitori puzzleUnlimited boards
- Daily hitoriA new board every day
- Solving a boardA 6×6, worked
- Hitori techniquesThe whole ladder
- Printed patternsTechnique — the opening
- Following throughTechnique — easy
- Corners and edgesTechnique — medium
- Sealed pocketsTechnique — hard
- Cutting the boardTechnique — expert
- Guessing at hitoriTechnique — refused
- Printable hitoriFor paper
- Hitori archiveEvery past board