I recently mentioned a problem of finding chains and necklaces in which adjacent links add to a square (or other things). Such problems can be restated as: Draw a graph having the numbers from 1 to n as vertices and an edge between any pair of vertices whose labels add to a square (or whatever). Is there a Hamilton path (circuit) in the graph? For example (requested by at least one listener), n = 32. 28 8 1 15 10 21 26 4 23 32 2 17 14 19 22 30 27 6 9 3 16 13 20 12 29 24 7 25 11 5 31 18 This can be upped to 33 by inserting ... 9 16 33 31 18 7 29 20 5 11 ... so we now have chains for n = 15 16 17 31 32 33 34 35 ... (and no smaller others??). This (and 32 33 ... for necklaces) seem to be in too much of an inchoate state for acceptance as sequences for OEIS ?? Finding Hamilton paths is NP-hard in general, but those of us who don't know about such things can often find them in particular cases. Any offers for contributions to knowledge, or references to existing knowledge that Ed Pegg hasn't told me about? R.