Symmetry-Based Enumeration of Polyominoes on C-Coloured Checkerboards

Polyominoes are shapes formed by joining edge-connected unit squares in the square lattice. We review their symmetry classification using dihedral group actions and enumerate their distinct, coloured variants, called “chrominoes”, which arise when polyominoes are placed on a C-coloured checkerboard. This colouring method, defined via modular arithmetic, assigns colours cyclically in both directions. Our main theoretical result establishes that for C=2, a polyomino has either one or two chrominoes, depending on whether it has a colour-reversing symmetry. Additionally, we introduce a new classification of two-colour chrominoes into 15 symmetry-based classes. For C≥3, the number of chrominoes is a small integer multiple of C, determined by the polyomino’s symmetry class and the symmetries of the infinite C-coloured checkerboard plane. These findings have applications to integer linear programming methods for polyomino tiling problems. We validate our theoretical results through exhaustive search in MATLAB and illustrate them with examples.