Suppose you have a checkerboard, and a set of dominoes. Each domino is twice the area of a square of the checkerboard. Clearly, you could cover the entire checkerboard with thirty-two dominoes. But here's the question: Suppose you chopped off two opposite corners of the checkerboard. Can you now completely cover the remainder of the board using thirty-one dominoes?
 

Solution

The answer to this question is: No, you cannot cover the checkerboard with 31 dominoes after two opposite corners have been removed.

But how to prove it? That's the question. The answer is amazingly simple.

If you are removing opposite corners, you are removing two squares of the same color. This leaves 32 squares of one color, and 30 squares of the other color. Since every domino must cover one square of each color, it is impossible to fully cover the checkerboard.

Isn't that slick?

1.

Can you take two corners from the same side of a checkerboard and cover the remaining squares with dominos?

2.

What if you take away any two adjacent squares?

3.

What if you take away two diagonally-touching squares?

4.

What if you took away all four corners?