Three cups problem
The three cups problem is a mathematical puzzle that, in its most common form, cannot be solved.
In the beginning position of the problem, one cup is upside-down and the other two are right-side up. The objective is to turn all cups right-side up in no more than six moves. You must turn over exactly two cups at each move.
Solution
Impossible version
The problem is impossible to solve. An even number of cups are facing up, and you must turn over exactly two cups at each move. Since an even plus an even is an even, not an odd, no number of even flips will ever get all the three cups right-side up. To solve the problem, you need an odd number (i.e., three) of cups facing up, so the problem is impossible.
Solvable version
The solvable (but trivial) version of this puzzle begins with one cup right-side up and two cups upside-down. To solve the puzzle in a single move, you need only turn up the two cups that are upside down — after which all three cups are facing up.