Regular pandiagonal sparse magic squares of order \(n\equiv 5 \pmod{6}\) with density 6
Bibliographic record
Abstract
Sparse magic squares are a generalization of magic squares and can be used to the magic labeling of graphs. An \(n\times n\) array based on \(\mathcal{X}\)\(=\{0,1,\cdots,nd\}\) is called a sparse magic square of order \(n\) with density \(d\) (\(d ), denoted by SMS\((n,d)\), if each non-zero element of \(\mathcal{X}\) occurs exactly once in the array, and its row-sums, column-sums and two main diagonal sums is the same. An SMS\((n,d)\) is called pandiagonal (or perfect) denoted by PSMS\((n,d)\), if the sum of all elements in each broken diagonal is the same. A PSMS\((n,d)\) is called regular if there are eactly \(d\) positive entries in each row, each column and each main diagonal. In this paper, some construction of regular pandigonal sparse magic squares is provided and it is proved that there exists a regular PSMS\((n,6)\) for all positive integer \(n\equiv 5 \pmod{6}\), \(n>6\).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.030 | 0.011 |
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 source (direct Gemma or distilled Codex), 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".