An Optimal Strategy for 2×n AB Games

Abstract

This paper presents new and systematic strategies for 2×n AB games. We invent a graphic model to represent the game-guessing process. From this representation, we find some symmetric and recursive structures in the process. This not only reduces the size of the search space but also helps us to derive the optimum strategies more efficiently. By using this novel approach, we develop optimal strategies for 2×n AB games in the expected and worst cases, and are able to derive the following new results: (1) n/2+1 guesses are necessary and sufficient for 2×n AB games in the worst case. (2) The minimum number of guesses required for 2×n AB games in the expected case is (4n3+21n2 -76n+72)/12n(n-1) if n is even, and is (4n3+21n2 -82n+105)/12n(n-1) if n is odd.