Tredoku is a generalization of Sudoku in which each 3×3 box forms part of the surface of a 3-dimensional stack of cubes. Like Sudoku, the goal is to fill the grid with the digits 1 up to 9 in such a way that each digit appears exactly once in every 3×3 box and in every 9×1 strip. The picture above shows a solved Tredoku grid with three of the 9×1 strips highlighted in red, green, and blue.
The recent paper Tredoku Patterns by Simon R. Blackburn investigates the possible shapes of Tredoku boards, such as the pattern shown above. A Tredoku pattern is a flat arrangement of rhombus-shaped tiles, but it should be thought of as a connected network of vertical and horizontal squares in three dimensions, where each tile in the pattern corresponds to a 3×3 box in a completed Tredoku puzzle. Notice that a Tredoku pattern cannot contain a run of four or more tiles stuck edge-to-edge in a straight line, because each 9×1 strip in the solved puzzle needs to correspond to a run of three tiles.
A Tredoku pattern is required to satisfy four conditions: (1) every run is of length 1 or 3; (2) the pattern is edge-connected; (3) the pattern is simply connected, meaning that it has no “holes”; and (4) the pattern remains connected after removing any one tile, even if it is only connected at corners. Here, edge-connected means that it is possible to walk from the interior of any tile to the interior of any other without passing through a vertex.
A leaf of a Tredoku pattern is a tile that is connected to the rest of the pattern via a single edge. The 13-tile pattern above contains 8 leaves, shown in green. Each of the non-leaf tiles is contained in exactly two runs of three tiles, and the pattern above contains a total of 6 runs of three tiles. It is a theorem that if ℓ, ρ, and τ are the numbers of leaves, runs, and tiles respectively, then we have ℓ=2τ−3ρ. Applying this formula to the picture above gives 8=(2×13)−(3×6).
The main result of Blackburn’s paper gives necessary and sufficient conditions for when a Tredoku pattern with parameters (τ, ρ, ℓ) exists. The possible values of τ and ℓ are shown by the shaded boxes in the picture above, and the values of ρ are determined from these by the formula ℓ=2τ−3ρ. Blackburn proves that if ℓ > ⌈τ/2⌉+1, or if (τ, ρ, ℓ)=(2ρ+1, ρ, ρ+2) for ρ ≥ 9, then no Tredoku pattern with parameters (τ, ρ, ℓ) exists. (Recall that the notation ⌈x⌉ refers to the smallest integer n satisfying n ≥ x.) There are also seven additional cases where no Tredoku pattern exists, including τ=12, ρ=8, ℓ=0, and τ=15, ρ=7, ℓ=9. Blackburn also proves that a Tredoku pattern with parameters (τ, ρ, ℓ) exists in all other cases, provided that we have τ ≥ 3, ℓ ≤ τ, and ℓ=2τ−3ρ. These results prove some conjectures of Donald A. Preece (1939–2014), who studied the problem in 2013.
The peak popularity of Tredoku seems to have occurred about 10 to 15 years ago, but it is still fun to play. If you would like to try a game for yourself, take a screenshot of the picture above. Tredoku reminds me of the 1980s video game Q*bert, which also took place on a stack of cubes (shown below).
Picture credits and relevant links
Tredoku is a trademark of Mindome Games.
The pictures of Tredoku puzzles are edited versions of pictures by Mark G. Watts.
The other graphics are edited versions of diagrams from the paper by Simon R. Blackburn.
The screenshot of Q*bert is by Spottedowl and appears on Wikipedia.
Substack management by Buzz & Hum.
I've never played Sudoku and didn't realize it was a logic game more than a math game. I can see how it could be addictive. Has Sudoku or Tredoku been explored for other number bases?