TITLE: An interesting puzzle
QUESTION [7 upvotes]: Here is a puzzle challenge for you:
Suppose $X,Y$ are independent and identically distributed Random Variables. Show that
$$P\{|X-Y|\le2\}\le 3P\{|X-Y|\le 1\}$$

REPLY [4 votes]: This is known as the 123 Theorem, and a proof can be found here.