Regarding the suggested equivalence between my "Most Wanted" 
Problem #1 and a problem related to Golomb rulers, Michael 
Brundage sent the following email:

Michael Brundage (brundage@ipac.caltech.edu) wrote:
> Your first problem mentions a relationship to Golomb rulers 
> (in the "Variations and Comments" section) on which I can 
> comment.
>
> First of all, Golomb rulers can be described as a problem in
> radioastronomy, where it is necessary to realize all the 
> distances 1, 2,..., N between pairs of satellite dishes 
> arranged on a line segment.  One (inefficient) solution is 
> to place a dish at each mark 0, 1, ..., N on a ruler.  Then 
> there is a pair (0, N) which are a distance N apart, and there 
> are two pairs (0, N-1) and (1, N) which are a distance N-1 
> apart, and so on.  The idea is to come up with more efficient 
> solutions, using as few dishes as possible.  For example, when 
> N = 6, the best solution uses only four dishes (or marks on 
> the ruler):  0, 1, 4, 6.  A Golomb ruler is a ruler which has 
> only these marks on it.
>
> Put another way, if ink were really expensive but you wanted 
> a ruler with which you could measure any integer distance (up 
> to some number N), then a Golomb ruler of length N would be the 
> most economical way to do it. A Golomb ruler of length 6 would 
> be one with marks at each end (0 and 6) and marks at 1 and 4 
> as well.
>
> Golomb rulers can also be described as graphs with a certain 
> labelling property (graceful graphs); these are described in 
> some detail at 
>
> Brundage on Graceful Graphs 
>
> (soon to be graceful/index.html, but I haven't had the time to 
> update these pages yet).  Each mark on the ruler corresponds 
> to a vertex in the graph; each time a pair of marks is used to 
> make a distance, the edge joining those vertices is used.  Each 
> vertex is labelled with the value of the mark; each edge is 
> labelled with the distance (which is the absolute value of the 
> difference of its endpoints' labels).
>
> For example, in the ruler of length six, we end up with the 
> complete graph on four vertices.  The vertices are labelled 
> 0, 1, 4, and 6, and the edges are labelled 1, 2, 3, 4, and 5.  
> The graph is the skeleton of a tetrahedron, roughly sketched 
> below (without the edge labels):
>
>                0
>               /|\
>              / | \
>             /  |  \
>             1--|--6
>              \ | /
>               \|/
>                4
>
> The relation to Problem 1, if I understand it correctly is 
> this:  There are two distinct sets of N collinear points such 
> that the set of N(N-1)/2 pairwise distances between them are 
> equal if (but not only if) there are two distinct graceful 
> labellings of the complete graph on N vertices.
>
> Unfortunately, the complete graphs on 5 or more vertices are 
> not graceful.  This does not imply that there are no two 
> distinct sets of N collinear points with the sets of pairwise 
> distances between them equal; however, it does imply that 
> if there are N such points (N>4) then the distances are not 
> consecutive integers.
>
> When N<4, nothing interesting is happening.  When N=4, we 
> have the graph above, which does have one other graceful 
> labelling, using 0, 2, 5, and 6.  (Every graceful graph has 
> a second graceful labelling; this is explained in the pages 
> I mentioned above.)  This means that the sets of points 
> {0, 1, 4, 6} and {0, 2, 5, 6} give rise to the same set of 
> distances (namely, {1, 2, 3, 4, 5, 6}).  (Presumably, this 
> is what you mean when you say that a point set and its 
> "reversal" both give rise to the same set of distances...)
>
> As you can see, Golomb rulers are different from your Problem 
> 1, and unfortunately nothing particular interesting happens 
> where the two overlap, mainly because in a typical Golomb ruler 
> only the distances between some of the pairs of points are 
> important, and there are only a few Golomb rulers for which 
> the distances between all the pairs of points (as in this 
> problem) are used.
>
> There is also a generalization of Golomb rulers to "Golomb 
> rectangles."  I haven't really looked into them at all, though 
> I know there was a paper by James Shearer published in the 
> second issue of the Electronic Journal of Combinatorics at
>
> Shearer on Golomb Rectangles 
>
> dealing with them.  It may be that something there is relevant, 
> though I doubt it (since then even more of the distances will 
> be ignored).