Arithmetic Billiard Paths Revisited: Escaping from a Rectangular Room
Bibliographic record
Abstract
In this work, we consider a problem where a robot can move in a straight line inside a 2D rectangular room with integer lengths until it hits any part of the wall of the room. If the robot hits any part of a wall other than the corners or any point of an opening, then the robot bounces off the wall and follows a new direction in another straight line following the laws of symmetric reflection. The robot needs to escape through an opening on the wall that has a minimum length of one unit. The robot can only escape through the opening if it reaches any point of the opening with a non-zero angle.We present an efficient algorithm for which the robot is guaranteed to find the opening if there is any or declare that there is none. We prove that the algorithm works if and only if the sides of the rectangle are co-prime. As a by-product of our main result, we also provide some interesting results related to the coverage of the interior of the rectangle when the robot follows similar algorithms to escape from the rectangular room.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".