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≤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
⌈4c/3⌉−1)-strategy for p=2 for all c≥2, and a ⌊(3c−1)/2⌋-strategy for p=3 for all c≥4 and show the optimality
of both strategies, i.e., we prove that no (k+1)-strategy for a smaller k
exists