I was just looking at Schur numbers, and sumfree sets. http://en.wikipedia.org/wiki/Sum-free_set The way it is defined there, a set is sumfree when no members a,b,c exist within the set such that a+b=c. I made the mistake of thinking that a,b,c had to be *distinct*. With this faulty reasoning, numbers 1-23 can be divided into 3 sumfree sets in 3 different ways. {{{1,2,4,8,11, 22},{3,5,6,7,19,21,23},{9,10,12,13,14,15,16,17,18,20}}, {{1,2,4,8,11,16,22},{3,5,6,7,19,21,23},{9,10,12,13,14,15, 17,18,20}}, {{1,2,4,8,11,17,22},{3,5,6,7,19,21,23},{9,10,12,13,14,15,16, 18,20}}} With the proper definition, numbers 1-13 can be divided into 3 sumfree sets. {{{1,4,7,10,13},{5,6, 8,9},{2,3, 11,12}}, {{1,4, 10,13},{5,6,7,8,9},{2,3, 11,12}}, {{1,4, 10,13},{5,6, 8,9},{2,3,7,11,12}}} Now a puzzle. Using my definition, where a+b=c does not occur with distinct a,b,c within the set, divide the numbers 1-8 into two sumfree sets. The answer is unique. Other reading: http://mathworld.wolfram.com/SchurNumber.html http://www.combinatorics.org/Volume_1/Abstracts/v1i1r8.html http://www.combinatorics.org/Volume_7/Abstracts/v7i1r32.html