Use the greedy algorithm to form a sequence of integer triples (1,2,3), (4,6,8), (7,10,13), (11,15,19), (12,17,22), (14,20,26), (9,16,23), (21,29,37), ... such that 1. the n-th triple is an arithmetic progression with common difference n; and 2. no integer is allowed to appear in more than one triple. Question: does the number 5 ever appear? (All numbers <5000 other than 5 do appear!)

(If you do the same thing for pairs, you get (1,2), (3,5), (4,7), (6,10), (8,13), (9, 15), (11, 18), (12, 20), ...; the n-th pair is ([n phi], [n phi^2]), where phi is the golden ratio.)
posted by Lionel Levine at 9:41 PM