Question
There is an 8x8 chess board in which two diagonally opposite squares have been cut off. You are given 31 dominos, and a single domino can cover exactly two squares.
Can you use the 31 dominos to cover the entire board? Prove your answer (by providing an example, or showing why it’s impossible).
Solution
The chess board initially has 32 black and 32 white squares. By removing opposite corners, we’re left with 30 of one color and 32 of the other color.
31 dominos must cover 31 of one color and 31 of the other color.
So, impossible.