Fifteen identical cups stand on a tray, every one of them upside down. A move consists of turning over exactly two cups, any two you like. Can every cup be made to stand the right way up?

Watch how the count of upright cups changes on a single move. Turning two cups that are both down raises it by 2, two that are both up lowers it by 2, and one of each leaves it unchanged. So the count changes by 2 or by 0 every time and can never switch between even and odd. It starts at 0, which is even, so it stays even forever, and 15 is odd. No number of moves can work. The traps all offer a count of moves, which is the wrong thing to think about — the barrier is not how many moves you spend but the odd-even property that no move can alter.