Two Shapes in the Numbers

These read the board as printed and need nothing else. Everything they have to say is available before you have made a single mark.

Twins side by side

Two identical numbers next to each other in a row or a column cannot both be painted out, because painted cells may not touch. So at least one of that pair survives — and if one of them survives, every other copy of that number in the same line has to go. Find the pair, then paint out every other copy along that line.

It is the single most productive move in the puzzle and the only one available on a completely untouched board, which is why hitori boards are built with repeated numbers packed close together. A line reading 4, 2, 4, 4, 6, 4 gives it away twice over: the touching pair at positions three and four settles both of the other 4s immediately.

Notice what it does not tell you: which of the pair survives. That is usually decided much later by something else entirely, and trying to work it out now is the classic way to waste ten minutes early in a board.

what the reasoning read what it decides what it wrote

A number caught between two of its own

Where a line reads x, then anything, then x again — the two matching numbers separated by exactly one cell — the cell in the middle survives, whatever number it happens to hold.

The argument takes a moment and then never leaves you. Suppose the middle cell were painted. Then both of its neighbours would have to survive, because painted cells cannot touch. But those two neighbours hold the same number and sit in the same line, and two identical survivors in one line is exactly what rule one forbids. So the middle cell cannot be painted, and it survives.

It is the first argument in this puzzle that is not simply a reading of the rules, and it is worth being able to reconstruct rather than merely remember, because the same shape of reasoning — suppose this, watch two rules collide — is what the later rungs are made of.

The run of three, which is both at once

Three identical numbers in a row are the case beginners meet first and it needs no rule of its own. The touching pair at the left end paints out the copy at the right; the touching pair at the right end paints out the copy at the left; and the sandwich argument keeps the middle. All three cells are settled from two ideas you already have.

That is the general shape of this rung. It contains exactly two arguments, both of them about the printed numbers, and they are worth doing once and thoroughly at the very start — because unlike everything above them on the ladder, nothing you subsequently do to the board will give them anything new to say.

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

More Hitori pages

• Hitori puzzleUnlimited boards
• Daily hitoriA new board every day
• Hitori rulesThree rules, in full
• Solving a boardA 6×6, worked
• Hitori techniquesThe whole ladder
• 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