The Parity Party Switches
A small parity puzzle where the trick is noticing what can never change, even when everyone is busy flipping switches.
Some operations change many things, but one parity fact stays fixed.
Puzzle
Try it before the answer.
There are 20 switches, all off. In one move, you must choose exactly two switches and flip both. After any number of moves, can you end with exactly one switch on?
Hints
- Do not track the switches one by one.
- Track whether the number of on switches is odd or even.
- What does flipping two switches do to parity?
Solution
No. The number of on switches starts even: zero. Flipping two switches changes the count by -2, 0, or +2, so the parity stays even. Exactly one on switch would be odd, so it is unreachable.
Watch What Cannot Change
The positions can change, but the odd/even count cannot. Track that invariant instead of following every switch.