The Mutilated chessboard
Question: Suppose a standard 8x8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares?
Answer: The puzzle is impossible. Any way you would place a domino would cover one white square and one black square. A group of 31 dominoes would cover 31 white and 31 black squares of an unmutilated chessboard, leaving one white and one black square uncovered. The directions had you remove diagonally opposite corner squares, and such squares are always either both black or both white.
The three cottage problem
Question: Suppose there are three cottages on a plane (or sphere) and each needs to be connected to the gas, water, and electric companies. Using a third dimension or sending any of the connections through another company or cottage are disallowed. Is there a way to make all nine connections without any of the lines crossing each other?
Answer: There is no correct solution: it is impossible to connect the three cottages with the three different utilities without at least one of the connections crossing another.
The three cup problem
Problem: Starting with three cups place one upside down and two right side up. The objective is to eventually turn all cups right side up in six moves. You must turn exactly two cups over each turn.
Answer: The puzzle is impossible. An even number of cups are facing up and you are allowed to turn two over at a time. Since an even plus an even is an even, not an odd, no number of even flips will ever get all the three cups face up. You need an odd number of cups facing up, so the problem is impossible. The possible version of this puzzle is to start with two cups facing down and one cup facing upward. This is possible. Turn up an even number (two) of cups, and all the cups are facing up; an odd plus an even is an odd (1+2 = 3).