On 11/26/06, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
Refering to http://www.mathpuzzle.com/AmbiguousTowns.gif
I made a mistake, which I didn't realize until after posting. Now that I understand the problem better, let me try rephrasing it. Actually, there are two different problems, one asked by Stan Wagon,
Between four towns are roads of length 3, 5, 6, 7, ~3.10, ~5.44 . There are two distinct town configurations. (See picture in link)
Q1. Are there 6 integer road lengths that lead to distinct town configurations?
Q2. Are there 10 road lengths that lead to distinct town configurations?
Ed Pegg Jr
I interpret problem Q1 as: Find an arrangement of 4 distinct points in the plane such that the 6 pairwise distances are distinct positive integers and such that by moving one point to a new location in the plane a new arrangement of 4 points results, not congruent to the first (by rotation or reflection), but with the same set of pairwise distances. I interpret problem Q2 as the same, except you start with 5 points. If this is the problem, then it is more a number theory problem than a geometry problem, because of the requirement of all distances to be integers. I tried the interesting construction of Bill Thurston (elsewhere in this thread) hoping to specialize it to Q1 and then force distances to be integers, but I always ended up with two equal distances. Perhaps I misunderstood something. So I abandoned that and tried my own path. The best I could do for Q1 was 4 of the 6 distances integers and the remaining two square roots of integers. Did anyone do better? Since, the planar arrangement is a tetrahedron with volume zero one might look for similar problems in 3-d. Here one finds that the volume of the 3-d tetrahedron is not uniquely determined by its edges. In fact there are as many as 30 different tetrahedra, with 30 different volumes and the same set of edges. The edges 7,8,9,10,11,12 give an example of this. Now find an integer example with two tetrahedra of volume zero :-). Jim Buddenhagen