This was a puzzle that my professor presented at the end of our discrete mathematics class as a problem to chew on for the weekend. I've been trying to crack at it this whole afternoon but haven't made any substantial progress, does anyone have any hints? I don't want to know the answer, just some guidance in the sort of direction I should go:
Alice and Bob alternately place a checker on an unoccupied square of an initially empty eight-by-eight checkerboard. The rule is that after Alice places her initial checker, every new checker must be orthogonally adjacent to the most recently placed checker. The players are in effect constructing a path of checkers. The last player to make a legal move wins the game. Your mission: find a winning strategy for Bob.
I tried playing games on a 4x4 and a 5x5, it's clear that Bob has an advantage in this situation but I'm failing to generalize anything specific from the games I've played. All I know is that if a game is played in which all the tiles of the board are covered in the 8x8, Bob will win, but pursuing in this direction seems too tedious to create a proof of the solution.
Bob colors the Board as follows, then he just makes sure that he always places a chip of the same color as the previous chip. Bob will never lose, because after Alice moves she always leaves the other square in the $2\times 1$ grid unnocupied.