Problem 1 asks for a proof that it's impossible to have two
incongruent arrangements of points on a straight line with the
same multi-set of distances.  Of course, straight line is a 
special kind of one-dimensional space, namely, open.  Torsten 
Sillke has pointed out that if we consider a CLOSED 1-D space, 
there are incongruent arrangemets with the same multi-set of 
distances, where each pair of points has two distances, measured 
along the loop in both directions.  (He actually just counts the 
shorter of these two "distances" in each case.)

At 08:57 PM 4/1/96 +0200, Torsten Sillke wrote:
> The smallest example of (non-congruent) sets with equal distance 
> spectra.  Take 4 points of the vertices of a regular 8-gon.
>
>      X -- X              . -- .      Both sets are symmetric
>     /      \            /      \
>    .        .          X        X
>    |        |          |        |
>    .        X          .        X
>     \      /            \      /
>      . -- X              . -- X
>
>  Distance Spectra:  1 2 3 4   (Distance measure is arc-length)
>             Count:  2 1 2 1

This "closed space" interpretation is very nice, and raises many 
interesting questions.  For example, it would be interesting to 
find incongruent arrangements of points on the surface of a sphere, 
such that their sets of point-to-point distances (measured in either 
direction along the connecting geodesics) were identical.

Also, Marc Le Brun (mlb@well.com) commented
> ...it suggests three different types of other possible answers:
>
>  1. Unbounded infinite subsets with *finite* distance multiplicities.
>  2. Bounded infinite subsets (say in the unit interval).
>  3. Finite (I personally doubt these exist).

In addition, Dan Hoey suggested that Problem 1 is a well-known open 
problem related to "Golomb rulers".  The terminology is apparently
not entirely standard, because Michael Brundage emailed a fairly 
detailed description of what he calls "Golomb rulers" and related 
problems (see Relation to Golomb Rulers) and says they 
are not the same as Problem 1.  Others have emailed me to say they 
focus on configurations with the same SET of distances, rather than 
the same MULTI-SET.  

By the way, the original problem can also be expressed as a
combinatorial problem on the integers.  Given a finite ordered
set of non-zero integers A = {a1, a2, ... ak}, let S{A} denote 
the multi-set of all the sums of consecutive elements of A.  
Also, define the "reversal" of A as A' = {ak, ... a2, a1}.  
Obviously S{A} = S{A'}.  Question:  Does there exist an ordered 
set B distinct from both A and A' such that S{B} = S{A}?