Alice and Bob have an infinite sheet of paper with a square grid on it. They play the following game: taking turns, they paint the sides of the squares (Alice is using the red colour, and Bob is using the blue colour); it is not allowed to paint any side which has been painted already. Alice was the first to make the move. Prove that Bob can play in such a way that Alice could never make a closed red contour.
I have been thinking about this question for several days, and I haven't found a way to answer it. I am assuming that there is an invariant/monovariant to exploit. The question was posed exactly as written above, on an undergraduate problem-solving sheet that I stumbled across online. Every other problem on the sheet could be solved quickly in a slick manner, so I'm presuming that this one can be too.
First, we cover the whole grid with triangles, as follows:
Now, every edge of the grid is in exactly one triangle. I propose the following strategy:
This strategy can be followed consistently. Now, Alice won’t be able to ever complete a closed contour, since if we could, of its lowest corners, we could take the leftmost one, and it would be incomplete. This completes our proof. $\blacksquare$