Tic-Tac-Toe on Designs
Bibliographic record
Abstract
We consider playing the game of Tic-Tac-Toe on block designs BIBD($v, k, λ$) and transversal designs TD($k, n$). Players take turns choosing points and the first player to complete a block wins the game. We show that triple systems, BIBD($v, 3, λ$), are a first player win if and only if $v \geq 5$. Further, we show that for $k = 2, 3$, TD($k, n$) is a first player win if and only if $n \geq k$. We also consider a weak version of the game, called Maker-Breaker, in which the second player wins if they can stop the first player from winning. In this case, we adapt known bounds for when either the first or second player can win on BIBD($v, k, 1$) and TD($k, n$), and show that for Maker-Breaker, BIBD($v, 4, 1$) is a first player win if and only if $v \geq 16$. We show that TD($4, 4$) is a second player win, and so the second player can force a draw in the regular game by playing the same strategy.
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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) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.018 | 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 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".