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.

If you get to 26, you are stuck. Assume 2 is red. Then,  6 must be blue, and 18 must be red. Now, 14 must be blue, since 2+2+14=18, and that makes 4 red, since 4+4+6=14.

What color is 26? Sincec 4+4+18=26 and 6+6+14=26,  we cannot color it either red or blue. Thus, if we get up to 26,  we will find x, y, z, and w (=26).

Posted by e.g. on 2006-12-06 11:17:47