Model the game as a minimax problem where the guesser selects guesses to minimize the maximum remaining candidate set size, and the adversary selects the response that maximizes it. Use a decision tree approach, choosing guesses that split the candidate set as evenly as possible, and provide pseudocode for the recursive minimax strategy.
Pro tip: Emphasize that the adversary's power is constrained by consistency with previous responses, so the guesser's optimal strategy is to maintain the set of all possible words and always guess a word that minimizes the worst-case remaining set size. This demonstrates understanding of game theory and algorithm design under adversarial conditions.
Define the game state as a set of possible target words consistent with all previous guesses and responses. The adversary can choose any word from this set and any response pattern that keeps at least one word consistent.
The guesser wants to minimize the maximum number of guesses needed in the worst case. This is equivalent to minimizing the depth of the decision tree, where each node is a guess and edges are adversary responses.
At each step, for each possible guess (from the candidate set or a larger dictionary), compute the partition of the candidate set based on all possible response patterns. Choose the guess that minimizes the size of the largest partition.
Implement a function that takes the candidate set and returns the optimal guess and worst-case number of guesses. Use memoization to avoid recomputing states.
Discuss the computational cost of evaluating all guesses and partitions, and potential heuristics (e.g., limiting guesses to candidate set) for practical implementation.
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