Three cups problem

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.

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.

[1] To see that the problem is insolvable (when starting with just one cup upside down), it suffices to concentrate on the number of cups the wrong way up.

Since a move inverts two cups and each inversion changes

by the sum of two odd numbers, which is even, completing the proof.

More generally, this argument shows that for any number of cups, it is impossible to reduce

is even, inverting cups two at a time will eventually result in