Suguru
The elegant polyomino number puzzle where no two identical digits can touch, even diagonally.
Basic Game Premise & Rules
MechanicsSuguru 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 board is divided into bold-outlined polyomino cages (regions) containing between 1 and 5 cells.
- Each cage of N cells must contain the integers from 1 up to N exactly once. A 1-cell cage contains only 1; a 2-cell cage holds 1 and 2; a 5-cell cage holds 1 through 5.
- King's Move Non-Adjacency: Two cells containing the same number can never touch each other in any direction—not horizontally, vertically, or diagonally at a corner.
- This non-adjacency rule applies everywhere across the grid, both within adjacent cages and across cage boundaries.
- Every puzzle has exactly one unique solution, deducible purely through logical elimination without guessing.
Rules by Example
See how each rule looks when done correctly versus when violated:
Numbers 1 to N in Each Region
Every region of N cells contains the numbers 1 to N exactly once. Numbers greater than N cannot be placed, and numbers cannot repeat within a region.
Equal Numbers Never Touch (Even Diagonally)
Equal numbers can never touch along a side or diagonally at a corner, whether in the same region or across different regions.
History & Origins
Origins & EvolutionSuguru was invented in Japan by Naoki Inaba, one of the modern era's most prolific and acclaimed puzzle creators. Inaba developed the format for publisher Nikoli, designing it to offer the accessible polyomino geometry of jigsaw puzzles combined with the rigorous deduction of Sudoku.
In the early 2010s, Dutch puzzle syndicate Keesing Media Group licensed and popularized the game across Europe under the name **Tectonic**. It rapidly captured widespread acclaim, becoming a daily staple in major international newspapers such as *The Guardian*, *The Daily Telegraph*, *De Telegraaf*, and *NRC Handelsblad*. Its appeal stems from its deceptively simple premise: unlike Sudoku, which relies on large 9×9 grids with fixed row and column constraints, Suguru operates on modular, organic clusters of varying sizes where diagonal exclusion creates deep local and global interactions.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Suguru formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Vertex Coloring on King's Graphs
Mathematically, Suguru can be formulated as a specialized vertex coloring problem on a King's graph G = (V, E). In G, vertices represent grid cells, and edges connect any two cells that are horizontally, vertically, or diagonally adjacent (the legal moves of a chess King). The board partition divides V into disjoint subsets C_1, C_2, ..., C_m. For each cage C_k, the assigned colors must form a bijection onto {1, 2, ..., |C_k|}. The diagonal non-adjacency condition enforces that G has a proper vertex coloring on the induced subgraph of each color.
Computational Complexity & NP-Completeness
When generalized to an n × n board with arbitrary polyomino cages, Suguru is NP-complete. The problem can be proved NP-complete via polynomial-time reduction from Planar 3-SAT or from Latin Square completion. Deciding whether a given partial assignment has a completion requires exponential time in the worst case under standard complexity assumptions (P != NP).
Constraint Satisfaction & SAT Reduction
Suguru is a classic finite-domain Constraint Satisfaction Problem (CSP). For each cell (r, c) and candidate digit d, a Boolean variable x_{r,c,d} indicates whether cell (r, c) holds value d. The rules translate directly to: (1) Exactly-one value per cell; (2) At-most-one appearance of each value per cage; and (3) Mutual exclusion clauses (¬x_{r,c,d} ∨ ¬x_{r',c',d}) for all Chebyshev-distance 1 pairs max(|r-r'|, |c-c'|) = 1. Modern CDCL SAT solvers and ILP solvers solve even large 20×20 grids in milliseconds using unit propagation and clause learning.
Polyomino Partition & Degree Constraints
The shapes of the cages (monominoes, dominoes, trominoes, tetrominoes, and pentaminoes) dictate the maximum degree and spatial reach of each digit. A cell with 8 King's neighbors eliminates candidate placements from up to 8 surrounding cells. When a 5 appears in a cage, its diagonal neighborhood immediately bounds the possible positions of 5s in neighboring cages, generating ripple-effect deductions across the grid lattice.
References & Further Reading
Bibliography- Inaba, N. (2008). Suguru Collection (Japanese: 数繰). Tokyo: Nikoli Publishing.
- Demaine, E. D., & Hearn, R. A. (2009). Games, Puzzles, and Computation. A K Peters / CRC Press. ISBN: 978-1568813226.
- Kendall, G., Parkes, A., & Spoerer, K. (2008). 'A Survey of NP-Complete Puzzles'. ICGA Journal, 31(1), 13–34.
- Keesing Media Group (2015). Tectonic: The Revolutionary Number Logic Puzzle. Keesing Publishing, London.
- Eppstein, D. (2012). 'Computational Complexity of Games and Puzzles'. University of California, Irvine.
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 | Gentle tutorials with small cages to master King's non-adjacency. | 3 |
| Easy | 5×5 to 6×6 | Clean deductions using cage boundaries and single candidates. | 12 |
| Medium | 7×7 to 8×8 | Inter-cage exclusions and pair eliminations. | 12 |
| Hard | 8×8 to 9×9 | Complex spatial pinching and diagonal chaining. | 12 |
| Expert | 9×9 to 10×10 | Master-tier puzzles requiring multi-cage lookahead. | 11 |
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