Question

You go to Mars and find a neighborhood with n Martians, n > 1. Say that two Martians are close if they live within one mile of each other. Show that there are at least two Martians who are close to the same number of Martians.

Answer 1

The relation "Close " is symmetric. And a Martian can't be "close " with itself. So a Martian can be close with at most n-1 Martians and at least 0 Martians.

Note that if we label each Martian and write down the "number" of Martians they're close with, the number 0 and n-1 can't occur at the same time in that sequence of n numbers .

Either way we can have only n-1 distinct numbers to attach to that n Martians. So clearly at least two Martians are close to the same number of Martians.