Seven cups all stand upside down. In one move you must turn over exactly two cups. Can all seven ever stand the right way up?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count how many cups face up after each move.
2The strategy
Track the number of upward-facing cups. Every move changes that number by two or by nothing at all, so ask what that does to whether the number is odd or even.
3The full solution
Start with 0 cups up, which is even. Each move changes the count by +2, 0 or -2, so the count stays even forever. Seven is odd, so it can never be reached.
Solution
The reliable way
Count the cups facing up. At the start that is 0, which is even. A move turns over two cups, so it either adds 2, subtracts 2, or leaves the count unchanged - in every case the count stays even. Since 7 is odd, the target can never be reached, no matter how many moves you make. The transferable idea: find a quantity no move can change, and use it to rule the target out.
The elegant way
The parity of the number of upright cups is invariant, and it starts even while the goal is odd, so no sequence of moves connects them.
Why this is on the test: Proving something impossible is a different skill from finding an answer, and parity is the first tool students get for it.