Symmetry-Based Enumeration of Polyominoes on C-Coloured Checkerboards
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".