Suppose you have a machine that can switch two people's bodies but each pair of bodies can enter the machine at most once. You and your friends keep switching bodies until everyone is in someone else's body. Is it possible to restore everyone to their original bodies? We'll be using MATH (in particular, cycles and permutations) to find out. The writers of the TV show "Futurama" actually proved that the answer is yes, using at most two extra people. *** Professor Farnsworth (in Bender's body): Now then Amy, we'll simply switch bodies, and then we'll...no, I'd be back in my body, but then you and Bender would be switched. And the Amy and Bender bodies can't trade minds again, since they just did! Amy (in Professor Farnsworth's body): Oh no! Is it possible to get everyone back to normal using four or more bodies? Professor Farnsworth: I'm not sure. I'm afraid we need to use...[dramatic music]...MATH!

Suppose you have a machine that can switch two people's bodies but each pair of bodies can enter the machine at most once. You and your friends keep switching bodies until everyone is in someone else's body. Is it possible to restore everyone to their original bodies? We'll be using MATH (in particular, cycles and permutations) to find out. The writers of the TV show "Futurama" actually proved that the answer is yes, using at most two extra people. If there's time, we will solve other silly improbable puzzles using cycles. *** Prof. Farnsworth (in Bender's body): Now then Amy, we'll simply switch bodies, and then we'll…no, I'd be back in my body, but then you and Bender would be switched. And the Amy and Bender bodies can't trade minds again, since they just did! Amy (in Prof. Farnsworth's body): Oh no! Is it possible to get everyone back to normal using four or more bodies? Prof. Farnsworth: I'm not sure. I'm afraid we need to use…[dramatic music] MATH!