Algorithm to cover maximal number of points with one circle of given radius

This is the "disk partial covering problem" in the literature -- that should give you a good place to start googling. Here's a paper covering one possible solution, but it is a little intense mathematically: Approximation Algorithms Design for Disk Partial Covering Problem

As a matter of fact, this falls in the area called computational geometry, which is fascinating but can be hard to get a toehold in. There's a good overview by deBerg on various algorithms related to the subject.

Edited to better wording, as suggested :

Basic observations :

• I assume the radius is one, since it doesn't change anything.
• given any two points, there exists at most two unit circles on which they lie.
• given a solution circle to your problem, you can move it until it contains two points of your set while keeping the same number of points of your set inside it.

The algorithm is then:

• For each pair of points, if their distance is < 2, compute the two unit circles C1 and C2 that pass through them.
• Compute the number of points of your set inside C1 and C2
• Take the max.