(In reply to Yet another approach.
Like Jer, I've applied the "reverse engineering" approach, trying to define what common feature is shared by the "reachable" triplets as opposed to "unreachable" ones.
Helas - I did not consider "going mod 3."
Comparing the 3 approaches presented so far I feel that to say that Paul's solution is much more elegant is an understatement.
It is the only general solution , valid for 3, 4, 5 triplet as well as for 333, 344, 338. (Just imagine finding all possible paths for the new triplet!)
Excellent proof, Paul!
Nice problem (I've rated it 4), Bractals!