Ed writes: << Refering to http://www.mathpuzzle.com/AmbiguousTowns.gif . . . 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?
Q1. Draw a non-isosceles triangle ABC with rational sides. Let M be the midpoint of AB, and let DD' be a segment drawn through M that's perpendicular to CM, such that the distances d(D,M) and d(D',M) are equal and rational. This implies distinct town configurations A,B,C,D and A,B,C,D' having the same sest of 6 distances. By scaling everything up by the product of all 6 denominators, the distances also become integral. Q2. Given an answer to Q1, add a 5th town on the perpendicular bisector of the segment DD', and far away from the other 4. This guarantees 10 road lengths with two distinct town configurations. It's unclear to me if this can be done with all distances integral. (Apropos this last comment, it's an unsolved problem whether there is a point in the plane at integral distances from all 4 vertices of an NxN square, for any integer N.) --Dan