Problem

Source: Benelux Mathematical Olympiad 2017, Problem 2

Tags: combinatorics, combinatorics proposed, games



Let n2 be an integer. Alice and Bob play a game concerning a country made of n islands. Exactly two of those n islands have a factory. Initially there is no bridge in the country. Alice and Bob take turns in the following way. In each turn, the player must build a bridge between two different islands I1 and I2 such that: I1 and I2 are not already connected by a bridge. at least one of the two islands I1 and I2 is connected by a series of bridges to an island with a factory (or has a factory itself). (Indeed, access to a factory is needed for the construction.) As soon as a player builds a bridge that makes it possible to go from one factory to the other, this player loses the game. (Indeed, it triggers an industrial battle between both factories.) If Alice starts, then determine (for each n2) who has a winning strategy. (Note: It is allowed to construct a bridge passing above another bridge.)