Rectangle-free four-colored square grids realized by cyclic-rotated reusable patterns
Why this work is in the frame
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Bibliographic record
Abstract
Abstract This paper deals with the very hard problem of grid coloring that has been specified more than 10 years ago in S. Fenner et al., Rectangle Free Coloring of Grids , 2009. Available at: http://www.cs.umd.edu/∼gasarch/papers/grid . pdf . It was unknown whether rectangle-free completely four-colored square grids of sizes 17 × 17 and 18 × 18 exist. As result of an extensive research I computed commonly with Christian Posthoff a rectangle-free pattern that can be reused cyclic-rotated by 90° for the four colors of the grid of size 18 × 18 (B. Steinbach and C. Posthoff, “Extremely complex 4-colored rectangle-free grids: solution of open multiple-valued problems,” in Proceedings of the IEEE 42nd International Symposium on Multiple-Valued Logic , Victoria, BC, Canada, ISMVL, 2012, pp. 37–44, B. Steinbach and C. Posthoff, “Rectangle-free colorings of extremely complex grids using 4 colors,” J. Mult.-Valued Log. Soft Comput. , vol. 21, nos. 5–6, pp. 609–625, 2013). A similar solution for the grid of size 17 × 17 has not been found so far. We explore in this paper for which sizes of square grids rectangle-free four-colored patterns exist, at which the pattern of one color can be reused for another color after a rotation by 90°.
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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 | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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 it