MathDB

Let

n\geq 2

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

I_1

and

I_2

such that:

\bullet

I_1

and

I_2

are not already connected by a bridge.

\bullet

at least one of the two islands

I_1

and

I_2

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

n\geq 2

) who has a winning strategy. (Note: It is allowed to construct a bridge passing above another bridge.)

Problem Statement