The Futurama Theorem

The Futurama Theorem:

I'm afraid we need to use math(s)!

Hypothesis: 

Bad news everybody...

Professor Farnsworth created a machine that allows the switching of the minds of two people. However, the switch can only occur between two people once only. There have been multiple switches between characters in the episode. 

There needs to be a maximum of two people who haven't previously used the machine in order to restore the original minds and bodies.

Proof:

(For those acquainted with group theory)

 First, let p be some K-cycle on [n] = {1…n}

Let (a,b) represent the transposition that switches the contents of a and b.

 Introduce two "new bodies" x and y to give p'

For any i = 1 to k, let s be the left to right series of switches

By verification: p's inverts the K-cycle and leaves x and y switched without performing the transposition (x,y)

 

By allowing p to be a perrmuation on [n], the sequence above shows the method of inversion. x and y can be switched manually after by (x,y) if need be.

QED

(In plain English)

1. Have everybody arrange themselves in circles, each facing their mind's original body
2. Get two new people (X and Y)
3. Fix the circles one by one
• Start each time with X and Y having their minds in their own body
• Unwrap any circle of people into a line 
• Swap the mind at the front into person X's body
• From the back of the line, have everybody swap with person Y in turn.
• Swap the mind in person X's body to the person at the back of the line
4. Swap the minds in the bodies of person X and person Y if need be

Hooray for Zoidberg!