Bibliographic record
Abstract
In this paper, we consider the Gergonne m-pile trick and its relation with the base m counting system. The case m = 3 produces one the oldest of mathematical �magic� tricks that involve the reordering of 27 cards. Joseph Diaz Gergonne [3], a French mathematician, was the first to analyze and generalize it in 1813. In [2, pag. 39], Gardner says: Mel Stover, of Winnipeg, Canada, calls my attention to the application of the ternary counting system to the Gergonne pile trick. Immediately, in [2, pag. 40], he also expresses: Reflecting on the above matters led Mr. Stover to the invention of a truly stupendous breath-taking version of the trick. It makes use of the decimal system and a deck of 10 billion playing cards! Based on these cases (m = 3 and m = 10), we demonstrate mathematically the existence of a formal relation between the position of the selected card after applying the Gergonne trick with a deck of mm cards and the base m counting system by using modular arithmetic. Also, we give general mathematical proofs of some particular situations as are: naming the position of the card, bringing the card to a named position and naming the card.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".