Bibliographic record
Abstract
Abstract The four river crossing problems in Alcuin’s Propositiones—numbers 17, 18, 19, and 20—deal with the fundamental nature of combinatorics, the mathematics of combinatorial structure. They are among Alcuin’s best-known problems. This chapter deals with these problems and their offshoots, including different cultural versions and more complex ones, such as the four-couple version. It also looks at combinatorics more generally, including problems in the field such as Kirkman’s Schoolgirls Problem. River crossing problems exemplify a recurring mathematical archetype—a problem that occurs in other languages and other eras, but with the same mathematical blueprint. The problems have a certain simple logic built into them that does not require any sophisticated training to understand. This may well have been Alcuin’s goal with these problems, showing that mathematical thinking is part of the human brain, manifesting itself in various cross-cultural ways. They are examples of how innate practical knowledge is transformed into theoretical knowledge by the brain, which is at the core of how mathematicians think.
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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 | 0.001 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 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".