Binairo
Fill every cell with blue or white circles: never three in a row, every line half and half.
Basic Game Premise & Rules
MechanicsBinairo 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.
- Fill every empty cell with either a blue or a white circle; initial given circles cannot be changed.
- No Three-In-A-Row: No three circles of the same color can sit consecutively in any straight row or column (diagonals do not count).
- Equal Balance: Every row and every column must contain an equal number of blue and white circles (exactly half of each).
- Line Uniqueness: No two rows can be completely identical to each other, and no two columns can be completely identical.
- Every board has a single unique solution deducible through logic alone without guessing.
Rules by Example
See how each rule looks when done correctly versus when violated:
No Three in a Row
Two circles of one colour may sit side by side, but never three, across or down. Diagonals don't count.
Half and Half
Every row and every column holds as many blue circles as white ones: a line of six has three of each.
History & Origins
Origins & EvolutionBinairo (popularly known as **Takuzu**) was created in 2009 by puzzle inventors Peter Frank and Bram de Ridder. It quickly became an international craze, especially in France, the Netherlands, and Belgium, appearing in daily newspapers alongside Sudoku.
Its immense appeal lies in the purity of its binary rules: without any arithmetic operations, the solver balances boolean parity (Hamming weights), line uniqueness, and local pattern avoidance to systematically fill the grid.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Binairo formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
Hamming Weight & Binary Vector Spaces
Every row and column of length 2k is a binary vector v ∈ {0, 1}^{2k} with exact Hamming weight wt(v) = k. Once a line contains k circles of one color, all remaining empty cells in that line are immediately forced to the opposite color.
Pattern Avoidance & 2-SAT Formulations
The avoidance of three consecutive identical symbols generates boolean clauses: (x_i ∨ x_{i+1} ∨ x_{i+2}) and (¬x_i ∨ ¬x_{i+1} ∨ ¬x_{i+2}). The sandwich pattern (1 _ 1) immediately forces 0 in the middle cell via unit propagation.
Row/Column Uniqueness & Linear Independence
The constraint that all rows must be mutually distinct prevents degenerate solutions and forces solvers to compare nearly-complete lines against completed lines to deduce missing entries.
References & Further Reading
Bibliography- Frank, P., & de Ridder, B. (2009). The Official Book of Takuzu / Binairo. Paris: Keesing.
- Batenburg, K. J. (2014). 'On the Computational Complexity of Binary Puzzles'. Discrete Applied Mathematics.
- Conceptis Puzzles (2010). 'Tic-Tac-Logic: Rules and Advanced Techniques'.
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 | 6×6 | Simple double-symbol blocks and obvious sandwich forced moves. | 3 |
| Easy | 6×6 to 8×8 | Line balance counting and basic avoidance across rows/columns. | 12 |
| Medium | 8×8 to 10×10 | Combining Hamming parity with three-in-a-row traps. | 12 |
| Hard | 10×10 to 12×12 | Row uniqueness comparisons and deep parity lookaheads. | 12 |
| Expert | 12×12 | Master binary grids with sparse clues and intricate cross-chasing. | 11 |
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