Mosaic
Shade cells guided by numbers that count how many cells around them are filled, revealing a hidden picture.
Basic Game Premise & Rules
MechanicsMosaic 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.
- Shade some cells to reveal a hidden pixel image.
- A number in a cell specifies exactly how many cells in its 3×3 neighborhood (its own cell and the 8 surrounding cells) are shaded.
- Cells on edges and corners have fewer neighbors (6 on an edge, 4 at a corner).
- A numbered cell may be shaded itself, in which case it counts toward its own clue.
- Cross out cells you deduce must stay white to systematically reveal the full mosaic.
Rules by Example
See how each rule looks when done correctly versus when violated:
Count the 3 × 3 Block
A number counts the shaded cells in its 3 × 3 block: its own cell and the eight around it. At an edge or a corner the block is smaller.
The Picture
Meet every number and the shaded cells draw a picture. Mark cells you know stay white.
History & Origins
Origins & EvolutionMosaic (also known globally as **Fill-a-Pix**) was invented in the late 1990s by British puzzle creator Trevor Truran and popularized worldwide by Conceptis Puzzles.
Inspired by the local neighborhood clues of Minesweeper, Mosaic eliminates the element of random clicking and guessing, converting the mechanic into a purely deductive, deterministic pencil puzzle. When the grid is completely solved, the black-and-white cells form a delightful, high-contrast pixel mosaic art piece.
Mathematical Concepts & Computational Complexity
Computer Science & Discrete MathBeyond its entertainment value, Mosaic formalizes core principles of discrete mathematics, theoretical computer science, and combinatorics:
2D Discrete Convolution & Cellular Automata
Mosaic is mathematically formulated as an inverse 2D convolution: given a binary image matrix X ∈ {0, 1}^{m × n}, the clue matrix C is obtained by convolving X with a 3×3 all-ones kernel K. Solving Mosaic is the inverse problem: de-convolving the observed discrete sums back into binary values.
System of Linear Inequalities over GF(2)
Each clue defines an exact sum equation ∑_{u ∈ N(v)} x_u = c_v over 0-1 variables. The system can be analyzed via Gaussian elimination and linear programming relaxations. When clues overlap, subtracting neighboring equations yields direct local assignments (e.g. difference of row blocks).
NP-Completeness via 3-SAT
Reconstructing a binary matrix from 3×3 neighborhood sums is NP-complete. Trevor Truran's format requires every valid puzzle to possess a unique solution reachable via deterministic steps.
References & Further Reading
Bibliography- Truran, T. (1998). Mosaic Collection. London.
- Conceptis Puzzles (2000). 'Fill-a-Pix: The Rules and Techniques'. Conceptis Ltd.
- Wolfram, S. (2002). A New Kind of Science. Wolfram Media. (Cellular automata in 2D lattices).
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 | Trivial 0s (all white) and 9s (all black) neighborhood starts. | 3 |
| Easy | 5×5 to 6×6 | Corner 4s, edge 6s, and adjacent difference reductions. | 12 |
| Medium | 7×7 to 8×8 | Overlapping 3×3 neighborhood subtractions revealing pixel art. | 12 |
| Hard | 8×8 to 10×10 | Multiple candidate shifts and dense image silhouettes. | 12 |
| Expert | 10×10 | Grandmaster mosaics requiring multi-block Gaussian elimination. | 11 |
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SeriesSlitherlink
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Place light beams in white cells so that every numbered black post sees its exact clue count.
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Place light bulbs to illuminate every white cell without any two shining on each other.
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A Latin square where every outlined cage satisfies its arithmetic target and operator.
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Place towers of heights 1 to N so that each outside clue sees the exact count of buildings.
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Partition the grid into rotational 180° symmetric galaxies, each centered on a single dot.
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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.
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Connect pairs of matching numbers with continuous non-crossing paths covering the grid.
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Connect the island network with straight horizontal and vertical bridges into one united world.
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Pitch a tent beside every tree with no two tents touching, matching row and column counts.
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Place circles and crosses in every cell without creating three in a row in any direction.