MathDB

Let

n\geq 3

be a positive integer. Alice and Bob are playing a game in which they take turns colouring the vertices of a regular

n

-gon. Alice plays the first move. Initially, no vertex is coloured. Both players start the game with

0

points.
In their turn, a player colours a vertex

V

which has not been coloured and gains

k

points where

k

is the number of already coloured neighbouring vertices of

V

. (Thus,

k

is either

0

1

2

.)
The game ends when all vertices have been coloured and the player with more points wins; if they have the same number of points, no one wins. Determine all

n\geq 3

for which Alice has a winning strategy and all

n\geq 3

for which Bob has a winning strategy.

Problem Statement