Suppose we have an infinite triangular lattice. In a competitive 2-person game, the players take turns colouring one point of the lattice which had not yet been coloured. There are 5 colours available. No point can be coloured a certain colour if one of the 6 adjacent points is that colour.
Player B has the goal of getting a point which can't be coloured - one which all 5 colours have already been used on adjacent points. Player A has the goal of preventing this - that is, ensuring that all points can be coloured. What strategy should each player play? Who will win?
My brother and I played on a non-infinite triangular lattice, and it seemed that generally Player A wins - although the optimal strategy wasn't clear, and maybe our board was just too small.