Optimal Strategies for Static Black-Peg AB Game with Two and Three Pegs

The AB~Game is a game similar to the popular game Mastermind. We study a version of this game called Static Black-Peg AB~Game. It is played by two players, the codemaker and the codebreaker. The codemaker creates a so-called secret by placing a color from a set of $c$ colors on each of $p \le c$ pegs, subject to the condition that every color is used at most once. The codebreaker tries to determine the secret by asking questions, where all questions are given at once and each question is a possible secret. As an answer the codemaker reveals the number of correctly placed colors for each of the questions. After that, the codebreaker only has one more try to determine the secret and thus to win the game. For given $p$ and $c$, our goal is to find the smallest number $k$ of questions the codebreaker needs to win, regardless of the secret, and the corresponding list of questions, called a $(k+1)$-strategy. We present a $\lceil 4c/3 \rceil-1)$-strategy for $p=2$ for all $c \ge 2$, and a $\lfloor (3c-1)/2 \rfloor$-strategy for $p=3$ for all $c \ge 4$ and show the optimality of both strategies, i.e., we prove that no $(k+1)$-strategy for a smaller $k$ exists.