Find the smallest integer n for which the following is possible:

Consider a set of positive integers {1,2...n}. Each number is painted with either red or blue paint, but no matter which numbers have which color, you can always find four (not necessarily different) numbers x, y, z and w that are coloured with the same paint and satisfy x+y+z=w.

I'm not claiming a minimum yet, but the minimum is 12 or less:

Assume that it can't be done with n = 12, and let 1 = Red
Then 3 must be Blue, because 1 + 1 + 1 = 3
Then 9 must be red, because 3 + 3 + 3 = 9
Then 4 must be Blue, because 1 + 4 + 4 = 9
Then 12 must be red, because 4 + 4 + 4 = 12
Then 2 must be blue, because 1 + 2 + 9 = 12
Then 8 must be red, because 2 + 3 + 3 = 8
Then 10 must be blue, because 1 + 1 + 8 = 10

But then 3 + 3 + 4 = 10 (All blue), so our initial assumption is wrong, and for n = 12 you can always find an x, y,z, w all the same color such that x + y + z = w

Posted by Steve Herman on 2006-12-06 13:02:24