Hitori
Shade the repeated numbers until every row and column has unique digits, keeping all white cells connected.
Basic Game Premise & Rules
MechanicsHitori is an untimed, deterministic pencil-and-paper logic puzzle played on a grid. Every valid puzzle has exactly one unique solution that can be discovered through rigorous deductive reasoning alone, without trial and error or guessing.
- Every cell in the grid contains a number. Shade some cells black so that no number appears twice in the unshaded cells of any row or column.
- No Adjacent Shading: Shaded black cells can never touch along a side (orthogonally), though they may touch diagonally at corners.
- Connected White Space: All unshaded (white) cells must remain connected into a single continuous piece through shared sides.
- Only Repeated Numbers Can Be Shaded: A shaded cell must contain a number that also appears unshaded elsewhere in its row or column.
- Every puzzle has exactly one unique solution discoverable purely through logical elimination.
Rules by Example
See how each rule looks when done correctly versus when violated:
No Repeats Among the White Cells
Shade cells until no number appears twice among the unshaded cells of a row or column. Shade only repeated numbers.
Shaded Apart, White Together
Shaded cells never share a side, and all the white cells stay joined in one piece. Circle a cell when you know it stays white.
History & Origins
Origins & EvolutionHitori was invented by Nikoli and first published in *Puzzle Communication Nikoli* #29 in March 1990. Its evocative full Japanese name, *Hitori ni Shite Kure* ('Leave Me Alone!'), whimsically captures the plight of the numbers: whenever duplicate numbers crowd into the same row or column, one must be eliminated (shaded) until every remaining number stands alone.
Hitori's genius lies in the inverted nature of its mechanics: rather than starting with a blank grid and adding numbers, the solver starts with a completely full number grid and whittles away redundant digits while ensuring the white landscape stays unbroken.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Hitori formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Independent Sets & Latin Square Reductions
The non-touching constraint for shaded cells mandates that the set of black cells forms an Independent Set in the grid graph. Simultaneously, the white cells must satisfy a Latin square-like property: each symbol appears at most once per line. This combines the maximum independent set problem with sub-permutation matrix extraction.
NP-Completeness via Planar Graphs
In 2009, Robert Hearn and Erik Demaine proved in *Games, Puzzles, and Computation* that Hitori is NP-complete. The proof constructs complex crossover gadgets and Boolean clauses where shading a number forces an avalanche of neighboring unshaded cells to preserve connectivity.
The Sandwich & Flanking Theorems
Simple Hitori deductions yield powerful invariant rules: (1) Sandwich Rule: If two identical numbers are separated by one cell (A B A), the middle cell B must be white; (2) Triplet Rule: If three identical numbers appear in a row (A A A), the outer two must be shaded and the middle kept white; (3) Diagonal Flank: If a cell is shaded, all four of its orthogonal neighbors are immediately guaranteed to stay white.
References & Further Reading
Bibliography- Hearn, R. A., & Demaine, E. D. (2009). Games, Puzzles, and Computation. CRC Press. ISBN: 978-1568813226.
- Nikoli (1990). Puzzle Communication Nikoli #29. Tokyo: Nikoli Publishing.
- Friedman, E. (2002). 'Pencil Puzzles: A Survey of NP-Completeness'. Stetson University Mathematics.
- Baste, J., & Vialette, S. (2014). 'On the parameterized complexity of Hitori'. Discrete Applied Mathematics.
Free Printable PDF Booklet
Download our beautifully formatted, high-resolution A4 puzzle booklet. Generated directly from the handcrafted levels on AZTest, it contains 50 puzzles ordered from gentle introductory boards up to master-level challenges, followed by complete solutions at the back.
| Difficulty Stage | Grid Size | Description | Puzzles |
|---|---|---|---|
| Intro | 5×5 | Introductory boards showing sandwich patterns and forced white circles. | 3 |
| Easy | 5×5 to 6×6 | Triplet eliminations and straightforward row/column uniqueness. | 12 |
| Medium | 7×7 to 8×8 | Diagonal black-cell corners and white connectivity preservation. | 12 |
| Hard | 8×8 to 9×9 | Multi-cell dead-end avoidance and long white wall tracking. | 12 |
| Expert | 9×9 to 10×10 | Master grids requiring intricate global parity and cut analysis. | 11 |
Explore More Logic Puzzles
SeriesSlitherlink
Connect dots to draw a single unbroken loop guided by surrounding cell numbers.
Wolves & Sheep
Draw a Slitherlink loop that fences the gentle sheep in and keeps the fierce wolves out.
Masyu
Draw one closed loop through cell centres that rigorously obeys the black and white pearls.
Nurikabe
Shade the sea to leave numbered white islands of exact sizes in one connected ocean.
Nonogram
Shade runs of cells according to row and column number clues to reveal a hidden picture.
Themed Nonogram
Solve color picture cross puzzles that blossom into vibrant pixel art upon completion.
Kurodoko
Shade cells so every numbered circle sees exactly that many white cells in its row and column.
Mosaic
Shade cells guided by numbers that count how many cells around them are filled, revealing a hidden picture.
Binairo
Fill every cell with blue or white circles: never three in a row, every line half and half.
Heyawake
Shade rooms according to their clues without isolating unshaded cells or leaving long straight corridors.
Searchlights
Place light beams in white cells so that every numbered black post sees its exact clue count.
Light Up
Place light bulbs to illuminate every white cell without any two shining on each other.
Calcudoku
A Latin square where every outlined cage satisfies its arithmetic target and operator.
Skyscrapers
Place towers of heights 1 to N so that each outside clue sees the exact count of buildings.
Sum Skyscrapers
Skyscrapers where each outside clue is the sum of the heights of all visible buildings.
Suguru
The elegant polyomino number puzzle where no two identical digits can touch, even diagonally.
Futoshiki
A Latin square puzzle with inequality signs pointing between neighboring numbers.
Fillomino
Divide the grid into polyomino blocks where each cell's number equals the area of its block.
Galaxies
Partition the grid into rotational 180° symmetric galaxies, each centered on a single dot.
Shikaku
Cut the whole grid into rectangles and squares, each containing exactly one number: its area.
L-Pieces
Draw borders to partition all white cells into perfect four-cell L-shapes.
Numberlink
Connect pairs of matching numbers with continuous non-crossing paths covering the grid.
Bridges
Connect the island network with straight horizontal and vertical bridges into one united world.
Slant
Draw a diagonal slash in every cell matching the corner numbers, without ever closing a loop.
Tents
Pitch a tent beside every tree with no two tents touching, matching row and column counts.
Marupeke
Place circles and crosses in every cell without creating three in a row in any direction.