Color each of the numbers 1 through n either red or blue such that if a+b=c then a, b and c are not all the same color. The addends are distinct.

For example with n=6 the sequence rbrbrb does not work because 2+4=6 but are all blue. Whereas rbrbbr does work.

What is the largest value of n for which such a sequence exists?

Note: Since the colors can be swapped, make the number 1 red.

Add a third color (green.) What is the new maximum value of n?

Putting in 2,8,23 in the integer encyclopedia found this problem is the basis of sequence A072842 

With four colors, the longest sequence is goes to 66 and with five the longest is 196

Posted by Brian Smith on 2009-10-14 20:59:44