Optimal Analyses for 3×n AB Games in the Worst Case

The past decades have witnessed a growing interest in research on deductive games such as Mastermind and AB game. Because of the complicated behavior of deductive games, tree-search approaches are often adopted to find their optimal strategies. In this paper, a generalized version of deductive games, called 3 × n AB games, is introduced. However, tranditional tree-search approaches are not appropriate for solving this problem since it can only solve instances with smaller n. For larger values of n, a systematic approach is necessary. Therefore, intensive analyses of playing 3×n AB games in the worst case optimally are conducted and a sophisticated method, called structural reduction, which aims at explaining the worst situation in this game is developed in the study. Furthermore, a worthwhile formula for calculating the optimal numbers of guesses required for arbitrary values of n is derived and proven finally.