Bibliographic record
Abstract
. We show that, in John Conway's board game Phutball (or Philosopher's Football), it is NP-complete to determine whether the current player has a move that immediately wins the game. In contrast, the similar problems of determining whether there is an immediately winning move in checkers, or a move that kings a man, are both solvable in polynomial time. 1. Introduction John Conway's game Phutball [1-3,13], also known as Philosopher's Football, starts with a single black stone (the ball) placed at the center intersection of a rectangular grid such as a Go board. Two players sit on opposite sides of the board and take turns. On each turn, a player may either place a single white stone (a man) on any vacant intersection, or perform a sequence of jumps. To jump, the ball must be adjacent to one or more men. It is moved in a straight line (orthogonal or diagonal) to the first vacant intersection beyond the men, and the men so jumped are immediately removed (Figure 1). If a jump is performed, the same player may continue jumping as long as the ball continues to be adjacent to at least one man, or may end the turn at any point. Jumps are not obligatory: one can choose to place a man instead of jumping. The game is over when a jump sequence ends on or over the edge of the board closest to the opponent (the opponent's goal line) at which point the player who performed the jumps wins. It is legal for a jump sequence to step onto but not over one's own goal line. One of the interesting properties of Phutball is that any move could be played by either player, the only partiality in the game being the rule for determining the winner. It is theoretically possible for a Phutball game to return to a previous position, so it may be necessary to add a loop-avoidance rule such as the one in Chess (three repetitions allow a player to claim a draw) or Go (certain repeated positions are disallowed). However, repetitions do not seem to come up much in actual practice.
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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 | 0.003 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.003 |
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; both teacher heads agree on what is shown here.
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".