Ed has pointed out what (in the cold light of day) should have been glaringly obvious: 5+8=13, and my example is a bit of a damp squib. I could modify the program to eliminate these; but a 5000 sec search up to sum of edges = 100 yielded only one other example, which suffers the same flaw. A triplet of tetrahedra with equal edges and the same volume turns out equally dubious. Heigh-ho --- back to the drawing-board! WFL On 11/27/06, Ed Pegg Jr <ed@mathpuzzle.com> wrote:

Very nice. This is the sort of configuration set I was looking for.

Unfortunately, in the first set, AB+BC=AC, so that part of it is a straight line, which is a slight flaw.

--Ed

Fred lunnon <fred.lunnon@gmail.com> wrote: Let the road lengths be arranged [AB, BC, CA, BD, AD, CD], where A,B,C,D denote cities. Then the follwoing two charts are distinct and planar, and both permute the same set of integers:

[5, 8, 13, 11, 9, 17] [5, 17, 13, 11, 9, 8]

[Given my capacity for wishful thinking, not to mention Maple's for wishful computation, I'd appreciate somebody verifying this independently!]

Fred Lunnon