The number of pessimistic guesses in Generalized Black-peg Mastermind

Mastermind is a famous two-player game, where the codemaker has to choose a secret code and the codebreaker has to guess it in as few questions as possible using information he receives from the codemaker after each guess. In Generalized Black-peg Mastermind for given arbitrary numbers p, c, the secret code consists of p pegs each having one of c   colors, and the received information consists only of a number of black pegs, where this number equals the number of pegs matching in the corresponding question and the secret code. Let b(p,c)b(p,c) be the pessimistic number of questions for Generalized Black-peg Mastermind. By a computer program we compute several values b(p,c)b(p,c). By introducing some auxiliary games and combining this program with theoretical methods, for arbitrary c   we obtain exact formulas for b(2,c)b(2,c), b(3,c)b(3,c) and b(4,c)b(4,c) and give upper and lower bounds for b(5,c)b(5,c) and a lower bound for b(6,c)b(6,c). Furthermore, for arbitrary p  , we present upper bounds for b(p,2)b(p,2), b(p,3)b(p,3) and b(p,4)b(p,4). Finally, we give bounds for the general case b(p,c)b(p,c). In particular, we improve an upper bound recently proved by Goodrich.

► We consider a version of Mastermind when the received information consists only of a number of black pegs. ► We generalize the game to an arbitrary number of pegs and colors. ► We give formulas for the pessimistic number of guesses in that game.