On Nov 29, 2006, at 3:52 PM, Michael Kleber 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]
I'm not sure why you decided it was somehow inferior, Fred! Yes, the first hexad has three cities collinear, but when handed a problem about distances along roads, nothing seems more likely than that some city is along the route between two others.
Did anyone verify this, per Fred's request? It certainly seems like the prettiest configuration seen so far. I drew this out, and it checks. It's an instance of the construction Dan first mentioned, triangle ABD with sides 5,11,9 is common to both, and vertex C is on the perpendicular at the foot of the median of this triangle from A to BD. Bill