Generalized Entropy and Solution Information for Measuring Puzzle Difficulty
Bibliographic record
Abstract
Metrics for problem difficulty are used by many puzzle generation algorithms, as well as by adaptive algorithms that want to provide players with the puzzles at the correct level of difficulty. A recently proposed general metric, puzzle entropy, combines an analysis of game mechanics with a model of player knowledge in the form of inference rules to predict problem difficulty. The entropy of a puzzle is the amount of information required, given a player’s knowledge about the puzzle, to describe a solution to a puzzle. This paper generalizes the concepts of puzzle entropy and solution information, providing a better foundation for the previous work and creating new algorithms, Minimum Solution Information and Total Solution Information. While functionally very similar to past work, the new algorithm allows knowledge about a puzzle to be represented as a policy, something that can be learned more easily. We then evaluate the impact of policies, inference rules, and player knowledge in the 2016 game, The Witness.
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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 | 0.003 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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".