Chess & Dominoes Puzzle

Maybe some of you have already seen these puzzles before, but for those who didn't, here are some fun brain twisters:

(1) Suppose you take an 8x8 chessboard, and you remove two diagonally opposite corner squares of the board (say a1, h8). Is it possible to cover the remaining 62 squares with 31 1x2 dominoes? If so, how, and if not, why not?

(2) What if we take an nxn board (where n is even), remove two opposite corners and try to do the same?

(3) And what if, instead of removing two squares in opposite corners, we remove two random squares of the 8x8 (or nxn) board? When can the remaining squares of such a board be covered with dominoes?

Seen the answers to 1 and 2 already.

No It can not

Is that right? and if it's right how many points do i get

I don't think we're doing points for this. Anyone can give a yes/no answer here without supplying proof and have a 50/50 chance of getting points, which is unfair. There is no point system here.

Please, if you reply, give a more elaborate answer then "Yes" or "No". Anyone can guess and get it right. That's also not what it's about: the proof is what matters and what makes the puzzle nice.

@ lid: I assume you're referring to the first puzzle, so please explain why not. And of course feel free to try (2) and (3) as well.

@ strangequark: You can also try problem (3), which is not so easy, even if you know the solutions to (1) and (2). Same for Eiwob: take a shot at (3) :)

I'm sure there are more ways to do this, but there's a nice proof in "Across The Board" (which I also mentioned in the books topic) which actually makes another connection with chess besides the chessboard.

Hint: Use a rook's tour...