The Magnus-Derek game

We introduce a new combinatorial game between two players: Magnus and Derek. Initially, a token is placed at position 0 on a round table with n positions. In each round of the game Magnus chooses the number of positions for the token to move, and Derek decides in which direction, + (clockwise) or (counterclockwise), the token will be moved. Magnus aims to maximize the total number of positions visited during the course of the game, while Derek aims to minimize this quantity. We define f (n) to be the eventual size of of the set of visited positions when both players play optimally. We prove a closed form expression for f (n) in terms of the prime factorization of n, and provide algorithmic strategies for Magnus and Derek to meet this bound. We note the relevance of the game for a mobile agent exploring a ring network with faulty sense of direction, and we pose variants of the game for future study.

• Type: Journal
• Year: 2008
• Where: TCS
• Authors: Zhivko Prodanov Nedev, S. Muthukrishnan
• Keywords: Derek, Game, Positions, TCS 2008, Theoretical Computer Science