Tredoku Pattern

A tredoku pattern is a finite arrangement of at least three lozenges, each formed from two adjacent triangles of an triangular grid, satisfying the conditions below. A run is a maximal sequence of tiles crossing successive shared parallel edges. Every run has length 1 or 3. The tiles are edge-connected, their union is simply connected, and deleting any one tile leaves a pathwise-connected region, allowing contact at corners (Blackburn 2026).

A leaf is a tile in exactly one three-tile run. If , , and count tiles, three-tile runs, and leaves, respectively, then

Blackburn (2026) proves that nonnegative integer parameters with , , and this identity are realizable precisely when apart from the exceptions below. Here is the ceiling function. The exceptional triples are , , , , , , and , together with all triples with . For example, leafless patterns exist with 9 tiles and with every multiple of 3 at least 15.

The patterns formalize a geometric question posed by Preece in talks in 2013 (Blackburn 2026), inspired by the Sudoku variant Tredoku. The classification concerns tile arrangements, not number assignments.