Three cups problem

The standard, unsolvable, arrangement of the three cups. As only one cup is upside down, turning it and another over leaves the other upside down.
The solvable version of the problem. Turning cups A and C over leaves all three upright.

The three cups problem, also known as the three cup challenge and other variants, 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, turning over exactly two cups at each move.

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, turn up the two cups that are upside down — after which all three cups are facing up. As a magic trick, a magician can perform the solvable version in a convoluted way, and then ask an audience member to solve the unsolvable version.

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