Determine the location of multiple static bodies based on their "gravitational" effect upon a dynamic point? I am writing a SF story, and though I'm sure that I've violated most of science and math to the Andromeda Galaxy and back, I'd like this part at least to be mathematically accurate. Here is a run down of the problem:
There's a group of objects known as 'the artifacts'. These artifacts have a mystical attraction to one another. This attraction functions along the same line as gravity, such that the attraction force (F) between any two artifacts is equal to $1/r^2$ (where r is the distance between the two artifacts). For this problem, all other forces acting on the artifacts can be ignored.
If the protagonists possess one of these artifacts and the means to measure the total force acting upon it, can they determine the position of the other artifacts (which can be considered to be unmoving)? The protagonists can reposition the artifact that they possess in between measurements.
How many measurements are needed if there are only 2 artifacts (including the one possessed by the protagonists)? 4 artifacts? 7? 8? Is there a general rule that tells you how many measurements are needed for N artifacts?

What I've done so far (which is basically just solving for the simplest system):
Variables
Known Artifact position -> ($X,Y,Z$)
Force on Known Artifact -> $F$
X-Component of Force on Known Artifact -> $F_X$
Y-Component of Force on Known Artifact -> $F_Y$
Z-Component of Force on Known Artifact -> $F_Z$
Unknown Artifact positions represented by ($X_1...X_N, Y_1...Y_N, Z_1...Z_N$)
Distance from Known Artifact to Artifact N -> $r_N$
Equations
$r_N = \sqrt{(X - X_N)^2 + (Y - Y_N)^2 + (Z - Z_N)^2}$
$F = 1/r^2$
$F_X = (X_N - X)/r_N^3$
$F_Y = (Y_N - Y)/r_N^3$
$F_Z = (Z_N - Z)/r_N^3$
Solution for System of 2 Artifacts
$r_1 = \sqrt{1/F}$
$X_1 = X + F_X\cdot r_1^3$
$Y_1 = Y + F_Y\cdot r_1^3$
$Y_1 = Z + F_Z\cdot r_1^3$

The system of 2 artifacts is trivial to solve, but the other solutions are more complex. Thank you in advance for your help. Also, is there a name for this type of math? I've been calling it reverse-trilateration.
 A: At risk of appalling the physicists, I will attempt to answer what I think is your question.
If the number of artifacts $N$ is unknown then the problem becomes quite hard. There are two cases: 


*

*You let the system evolve, so that all artifacts, yours and theirs, move around. In this case you could compare the evolution with the equations of motion predicted to solve for the number and position of the artifacts. However, $n$-body problems such as this do not admit closed-form solutions: see Wikipedia.

*You simply move the artifact around and take measurements, assuming no change in the other artifacts. In this case, the "gravitational attraction" will act as if all the artifacts were gathered at their center of gravity. For example, a tennis ball falling to Earth acts as though one giant "Earth entity" at the center of the planet is pulling it in. There is no way of telling how many particles are in the Earth or where they are, based solely on that information.
If the number of artifacts $N$ is known then, "in theory", there may be a solution.
You have $3N$ variables to determine, so you would need to take $3N$ measurements. Then you would get a giant non-linear system of equations which almost certainly would not be exactly solvable (but maybe in your book that's different!). 
Any edits/corrections from offended physicists are more than welcome :)
