$n\times n$ Hexagonal Grid problem

Question

Imagine an $n\times n$ hexagonal grid in parallelogram form that one pair of opposite sides are green and the other are blue. Is it possible to create a grid such that you can fill in only $n^2/2 + 0.5$ hexagonal greens and $n^2/2 - 0.5$ hexagonal blues and there is no path joining a blue side to a blue side nor a green side to a green side using the same color?

Answer 1

Not possible. Since the two numbers of colors differ by 1, the finished board is an example of the game Hex, and it's known that no such game can end in a tie.

See here for the game, and it was Nash who gave one proof of no tie.

Answer 2

Hint:

• Start in one of the corners and try to make a path along edges of the tiles, a path that has at all times, consistently, green color on one side (whatever it is, colored tile or side) and blue on the other.
• Prove that you can tell from where the path ended which pair of sides is connected.
• Is has to end somewhere, so one of the pairs has to be connected.

Some other info:

• This is the setting of the game of hex.
• Observe, that the theorem would not be true if the tiles were squares (e.g. colored like a chess board).

I hope this helps $\ddot\smile$