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.

Following your red/blue method, if 2, 4, and 18 are red, then 6, 8, 10, 12, and 14 must be blue, and that produces another conflict: 24 can be 2+4+18 or 6+8+10: we find all different x,y,z,w.

Posted by Old Original Oskar! on 2006-12-06 11:45:50