Ordering 15 marbles with a three-way scale
Why this work is in the frame
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Bibliographic record
Abstract
Many mathematical puzzles involve using a scale to compare the weights of two objects, or two groups of objects. The present work studies a similar but different type of question involving a scale.We are given a set of marbles, whose weights are all different. The only way to distinguish them is to use a special kind of scale. The scale has three trays and each can accept exactly one marble. The scale then indicates which is the heaviest, the lightest and consequently the middle one of the three marbles.The paper studies the question of ordering 15 different marbles with as few weighings as possible. The problem is taken from the website Enigmes, casse-têtes, curiosités et autres bizarreries [1] and Toppuzzle [2] where the best reported strategy requires 23 weighings. The problem can also be found at the website Trick of Mind [3] where a strategy requiring 22 weighings is proposed. These websites contain many challenging mathematical puzzles, some are classical problems, others are original creations. The author is not aware of any other work on this problem.The present paper describes a strategy that improves on the value of 22 weighings, and is structured as follows. Section 2 introduces some useful definitions and provides strategies to sort up to 9 marbles. Section 3 proves our main result, that 20 weighings is always sufficient to sort 15 marbles.
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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 | 0.001 | 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 | 0.001 | 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