Dominosa

Dominosa is a logic puzzle where a rectangular grid is filled with numbers, and the player must partition it into dominoes — pairs of adjacent cells (horizontal or vertical). Each possible domino combination from a standard set (e.g., 0-0 through 6-6 for a size-6 puzzle) must appear exactly once. The challenge is figuring out where to draw the borders between domino pairs so that every combination is used and no combination is repeated.

Rules

1. The grid is pre-filled with numbers ranging from 0 to N (for a size-N puzzle).
2. The grid dimensions are (N+1) x (N+2), containing exactly enough cells for all domino pairs from 0-0 to N-N.
3. Divide the entire grid into dominoes — pairs of two horizontally or vertically adjacent cells.
4. Each domino combination (e.g., 2-5, 3-3, 0-6) must appear exactly once.
5. Every cell must belong to exactly one domino.

Strategies

• Unique Pairing: If a particular number combination (e.g., 4-6) can only be formed in one place on the grid, that domino placement is forced.
• Elimination by Counting: Track which domino combinations have been placed. As more dominoes are fixed, remaining combinations become easier to locate.
• Forced Neighbors: If a cell's number can only pair with one adjacent cell to form a still-available combination, that pairing is forced.
• Chain Deduction: Placing one domino may leave a neighbor with only one valid pairing option, creating a chain of forced placements.
• Parity and Connectivity: In tight areas of the grid, consider whether remaining cells can be evenly divided into dominoes. An odd number of unassigned cells in a closed area signals an error.
• Cross-referencing Rows and Columns: If a combination like 3-5 appears adjacent in multiple places, check which placements are compatible with surrounding constraints.

Dominosa derives its name from dominoes, the classic tile game that has existed for centuries. The puzzle was first formulated as a mathematical recreation in the 19th century. The core idea — tiling a grid with a complete set of dominoes — was studied by mathematicians interested in combinatorics and tiling problems.

The puzzle appeared in recreational mathematics publications throughout the 20th century, including works by Martin Gardner in his famous "Mathematical Games" column in Scientific American. Gardner introduced many readers to the elegance of domino tiling problems and their connections to graph theory and combinatorics.

Dominosa gained popularity as a standalone logic puzzle in the 1990s and 2000s through puzzle magazines and early puzzle websites. Simon Tatham included it in his well-known Portable Puzzle Collection, which helped bring it to a wider audience. The puzzle has since appeared in various puzzle apps and books.

The mathematical underpinnings of Dominosa are well studied. The problem is equivalent to finding a perfect matching in a graph, and the uniqueness of solutions connects to deep results in combinatorial mathematics. Despite this theoretical depth, the puzzle remains accessible — anyone who can count and compare numbers can learn to solve it.