What is the maximum number of particles that can be arranged so that each one touches every other one?

Question

What is the maximum value of $n$ for which we can arrange $n$ particles in such a way that each particle touches every $n-1$ remaining particles in 3-dimensional space? A "particle" is a sphere of extremely small size. My friend and I think that the set of values of $n$ satisfying this condition is $\{1,2,3,4\}$. Are we correct or wrong?

Answer 1

If I understand you correctly, you have n spheres with each known radius r.

Now if you want to make all n spheres touch, look at (n-1)-dimensional triangles with edge length 2r and the spheres on the vertices.

By that I mean, for n=1, you just have a lone sphere, touching nothing.
For n=2, imagine a line that is connecting both spheres.
For n=3, imagine a triangle where each sphere is connecting to the other two.
Now, for n=4, imagine a tetrahedral, where still every sphere touches each other.

For n=5, you would need a 4-dimensional space, but if I understand correctly, you only have 3 dimensions to work with, so n=4 would be your maximum.