Suppose you select 501 numbers from 1 to 1000, inclusive. Show that if no two of the selected numbers differ by 334, then some two of the selected numbers differ by 666.
For each number N you pick, reject N-666 or N+334 -- one of them will be between 1 and 1000, and the other won't.
For the first 500 numbers you pick, you'll reject other 500.
But when you pick the 501st, you'll necessarily pick a rejected one.
Solution
