Shikaku
Cut the whole grid into rectangles and squares, each containing exactly one number: its area.
Basic Game Premise & Rules
MechanicsShikaku 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.
- Draw borders along grid lines to partition the entire grid into rectangles and squares.
- Every rectangle must contain exactly one number.
- The number in each rectangle specifies its exact area (how many cells it covers). For example, a 6 can be 1×6, 2×3, 3×2, or 6×1.
- Rectangles cannot overlap, and no cell can be left unassigned.
- Every board has a single unique rectangular partition discoverable purely through factorization and spatial deduction.
Rules by Example
See how each rule looks when done correctly versus when violated:
One Number, Its Area
Every rectangle holds exactly one number, and the number is how many cells the rectangle covers.
Rectangles Only
Every region is a rectangle (a square counts), and together they cover the whole grid. A 4 can be 1 × 4, 2 × 2 or 4 × 1, but never an L.
History & Origins
Origins & EvolutionShikaku was invented by Nikoli and debuted in *Puzzle Communication Nikoli* #34 in 1991 under the title *Shikaku ni Kire* ('Cut into Rectangles').
It is one of the most accessible and pedagogically rich pencil puzzles ever conceived: solvers learn to decompose numbers into integer factor pairs (such as 12 = 1×12, 2×6, 3×4, 4×3, 6×2, 12×1) while fitting them into geometric space. Its widespread adoption in schools and puzzle magazines reflects its perfect blend of elementary arithmetic and spatial planning.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Shikaku formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Exact Cover by Rectilinear Rectangles
Shikaku is an applied instance of the Exact Cover Problem: given a universe U of grid cells, select a sub-collection of candidate rectangles R_1, ..., R_k such that each cell is covered exactly once and each clue is matched. Donald Knuth's Algorithm X (Dancing Links / DLX) provides a canonical solving engine for Shikaku.
NP-Completeness via Planar 3-SAT
In 2008, Erik Demaine, Robert Hearn, and Michael Hoffmann proved that Shikaku is NP-complete. Even when rectangles are restricted to areas of small primes and composites, orchestrating choices of orientations can encode Boolean SAT formulas.
Prime Areas & Corner Anchors
Prime numbers P (such as 2, 3, 5, 7) have only 1×P or P×1 orientations. If a prime clue sits near a corner or boundary, its orientation is often immediately forced. Furthermore, if two clues share a candidate rectangle, inclusion-exclusion eliminates overlapping territory.
References & Further Reading
Bibliography- Demaine, E. D., Hearn, R. A., & Hoffmann, M. (2008). 'Shikaku is NP-complete'. Workshop on Combinatorial Algorithms.
- Nikoli (1991). Puzzle Communication Nikoli #34. Tokyo: Nikoli Publishing.
- Knuth, D. E. (2000). 'Dancing Links'. Millennial Perspectives in Computer Science.
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 | Corner primes and small squares demonstrating rectangular tiling. | 3 |
| Easy | 5×5 to 6×6 | Clean factorizations: 4s (2×2, 1×4) and 6s (2×3). | 12 |
| Medium | 7×7 to 8×8 | Overlapping factor candidates and edge boundary traps. | 12 |
| Hard | 8×8 to 10×10 | Large composite areas (8s, 10s, 12s) requiring multi-cell elimination. | 12 |
| Expert | 10×10 | Master grids with sparse clues and intricate interlocking rectangles. | 11 |
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