How to Solve a Hitori Puzzle

One real 6×6, from an untouched grid to a finished region. Every move below is the move the solver made, in the order it made it.

Patterns first, cascades second, shapes last

Start by reading the numbers and nothing else. Look along every row and every column for two identical numbers side by side, and for a number caught between two copies of another. Both settle cells outright, both are available before you have touched anything, and neither will ever tell you more than it does on this first pass — so do it once, thoroughly.

Then follow the consequences, and keep following them. Every cell you paint keeps its four neighbours. Every cell you keep paints out its twins along both of its lines. Every one of those decisions is another cell to follow from, and on an easy board this chain reaction is the entire solve. The people who find hitori frustrating are almost always the ones who make a decision and then go looking somewhere else instead of chasing it.

Only when the chain runs dry is it worth thinking about shape. Look at the corners first, because a corner cell has two ways out and runs short of them long before anything in the middle does. Then look for necks — places where the surviving region is one cell wide — and ask what would happen if that cell were painted.

One 6×6, from the first glance to the last cell

The board below is a 6×6 graded expert, chosen not because a beginner should start there but because its solve happens to use all five arguments on the ladder, in order, and because the last three arrive one after another at the end. The five positions shown are the five moments where something new is being used; the nineteen moves in between are more of the same.

Read each pair of boards left to right. On the left is the position as it stood, with the cells the argument read marked and the cell it decides outlined. On the right is the same position a moment later. A cell drawn dark has been painted out, a cell drawn pale has been kept, and a cell drawn plain is one nobody has decided yet — which on this puzzle is a real third state rather than an absence, and the reason the pictures are legible at all.

1. Before anything is painted, read the printed numbers. The first column holds a 3 at row 3 and a 3 at row 5, with one cell between them — so that cell survives, because painting it would force both 3s to stay and no line may keep two of the same number.

   The position

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   A moment later

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   Printed patterns

2. Now chase it. The cell just kept holds a 5, so no other 5 in its column may survive alongside it, and the 5 at the top of that column is painted out. That painting will in turn keep all four of its neighbours, and the chain runs on from there.

   The position

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   A moment later

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   What a decision forces

3. Nineteen moves of the same two ideas later the board stops. Look at the bottom-left corner: painting the 2 at row 4, column 2 would wall four numbers into it with nothing but painted cells in the way, which no answer can do — so the 2 survives.

   The position

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   A moment later

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   Corners and edges

4. The same argument again, one size up and no longer a shape you recognise. Painting the 3 at row 3, column 5 would seal five numbers into the top-right of the board, and the only way to be sure of that is to cover the cell and walk the region.

   The position

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   A moment later

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   Sealed pockets

5. And once more at full stretch. The 3 at row 4, column 4 is the last gap between two diagonal chains of painted cells, and painting it would leave eleven survivors on one side and fourteen on the other with no route between them.

   The position

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   A moment later

   	1	2	3	4	5	6
   1	5	4	6	5	2	2
   2	2	5	4	4	6	3
   3	3	1	4	4	3	5
   4	5	2	6	3	1	2
   5	3	6	1	2	4	1
   6	4	1	1	6	4	2

   Cutting the board

what the reasoning read what it decides what it wrote

Nineteen more moves of the same five ideas, and every cell is decided. Here is the finished board.

When the board gives you nothing at all

Chase your last decision again before concluding anything. By a wide margin the commonest cause of a stuck hitori board is a cell decided several moves ago whose consequences were never followed — a survivor whose twins are still sitting there undecided, or a painted cell whose neighbours were never marked as kept. Both are free to fix and both usually set off a run of two or three more.

If that comes up empty, walk the four corners and then the edges. Ask of each surviving cell how many of its neighbours are still unpainted, and if the answer is one, that neighbour survives too. This is a fast sweep and it is where a board at this level most often gives way, because the edges of the grid are where the surviving region runs out of room.

And if both of those fail, trace the surviving region and look for the narrow places. Somewhere there is a cell that everything on one side has to pass through, and painting it would leave that side stranded — so it survives. What the board never wants is a guess: every board published here was checked to be finishable without one.

Common questions

More Hitori pages

• Hitori puzzleUnlimited boards
• Daily hitoriA new board every day
• Hitori rulesThree rules, in full
• 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