7-8 grade You have a set of 32 domino pieces and the chess board. It is easy to show that you can cover the board with all pieces so that there is no overlap between the pieces and the whole board is covered. Now take the main diagonal, and remove 2 corner squares on it from the board, also discard one domino piece. Can the board still be covered with the remaining 31 pieces?
Answer
Notice that the domino piece always covers one black and one one white spots. The diagonally opposite spots on the board are the same color, hence you won't be able to cover a board like this with the domino pieces.
Problems like this are the essence of the oral exams in Math to enter Moscow University.
Most importantly, one has to think "out of the box" and not get scared at the very sight of the problem. The problems tend to reduce to checking a few constraints but finding those could be a trick on its own. My experience is that the more you do them the more trained you become, so keep on doing them or post the tricky problems you are aware of here.
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