Nonogram
Shade runs of cells according to row and column number clues to reveal a hidden picture.
Basic Game Premise & Rules
MechanicsNonogram 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.
- The numbers beside each row and above each column specify the lengths of consecutive shaded blocks in that line, in exact order.
- Every run of shaded cells must be separated from adjacent runs in the same line by at least one unshaded (white) cell.
- Fill in cells that must be shaded, and cross out cells you know must stay white.
- When all numbers in every row and column are satisfied, the completed grid forms a recognizable silhouette or pixel picture.
- Every puzzle has a unique solution discoverable purely through logical line deductions.
History & Origins
Origins & EvolutionNonograms were invented independently in 1987 by Japanese graphics editor Non Ishida (who created 'Window Art' pictures by turning on skyscraper windows) and professional Japanese puzzle creator Tetsuya Nishio.
Ishida published them in Japan under the name **Nonogram** (a blend of 'Non' and 'diagram'). In 1990, James Dalgety in the UK syndicated them in *The Sunday Telegraph*. In 1995, Nintendo released *Mario's Picross* on the Game Boy, cementing the puzzle as a global video game and pop-culture phenomenon. It remains one of the most widely played picture-logic puzzles in human history.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Nonogram formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Discrete Tomography & Reconstruction
Nonograms are an applied instance of Discrete Tomography: reconstructing an unknown binary image matrix from its projected 1D line sums and run lengths. In medical imaging and crystallography, similar problems arise when reconstructing atomic lattice structures from X-ray projections.
NP-Completeness via 3-SAT
In 1996, Nobuhisa Ueda and Tadaaki Nagao proved in *Information Processing Letters* that 2D Nonogram reconstruction is NP-complete. Even when every row and column has at most two runs, determining whether a satisfying grid exists requires exponential time in the worst case.
1D Overlap & Dynamic Programming
Single-line Nonogram solving can be completed in polynomial time via dynamic programming or regular expression parsing. When a line of length L has clues (c_1, ..., c_k) whose sum plus gaps leaves slack S = L - (Σ c_i + k - 1), any clue c_i > S must contain an overlapping core of c_i - S forced shaded cells.
References & Further Reading
Bibliography- Ueda, N., & Nagao, T. (1996). 'NP-completeness Results for NONOGRAM via Parsimonious Reductions'. Technical Report, Tokyo Institute of Technology.
- Batenburg, K. J., & Kosters, W. A. (2009). 'Solving Nonograms by combining heuristics with integer programming'. Discrete Applied Mathematics, 157(13), 2969–2983.
- Ishida, N. (1988). Nonogram Collection. Tokyo: Bungeishunju.
- Dalgety, J. (1990). 'The Sunday Telegraph Nonograms Book'. London.
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 | Simple single-run rows and overlapping core demonstrations. | 3 |
| Easy | 5×5 to 7×7 | Multiple runs, cross-outs, and basic edge overlaps. | 12 |
| Medium | 8×8 to 10×10 | Interlocking row-column deductions revealing pixel shapes. | 12 |
| Hard | 10×10 to 12×12 | Small run gaps, contradictory placements, and edge anchoring. | 12 |
| Expert | 12×12 to 15×15 | Dense, intricate pictures requiring multi-line lookahead. | 11 |
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