Chess & Dominos: a puzzle

Suppose you have a chess board that lacks two of its corners, one each from opposite corners of the board, like this:

Next to the chess board, you have thirty-one standard dominoes, each equal in size to two neighboring squares in the chessboard. These dominoes may be oriented in any of four ways:

or or or

Can you cover this chess board with domino pieces in such a way that each square on the board is covered by exactly one half of a domino piece (pieces should not exceed the border of the board)? Note that it is not required that the colors match. I merely colored the pieces to make them easier to see. (But further note, that if you laid a domino such that both colors were wrong, you could merely rotate it 180 degrees, and it would match the colors).

Okay, your turn, can you solve the puzzle. Feel free to take a break, get out your chess board and dominoes set, and test it out. Then come back here.

Okay, I see you're back. I gave you a VERY BIG hint when I colored the chess board and the dominoes the same color, and when I pointed out that it is possible to rotate domino pieces so that the colors would always match.

By now, you have not yet found an answer, so let me rewrite the puzzle slightly.

Can you break the 31 dominoes into halves, 31 red squares and 31 blue squares, and cover the chess board with red domino halves covering red chessboard squares, and blue domino halves covering blue chessboard squares?

Once again, I see you've come back and told me you can't do it.

Now, let me ask you some questions?

How many red squares are on the chess board
How many blue squares are on the chess square
How many red squares are on your supply of 31 domino pieces
How many blue squares are on your supply of 31 domino pieces

You now have enough information to answer the question:

Is it possible to cover this 62-square chessboard with exactly 31 dominoes?

Obviously, you cannot do so. You can place 30 pieces on the chess board, but you will end up with two red squares left over on the chess board, and one domino. You cannot cover the chess board according to the rules of either the original problem or the revised problem.