Four identical spheres (like the ones shown in blue in the above cube case) are arranged in a pyramid, such that each sphere is tangent to the other three. If the radius of the four spheres is R, what is the radius r of the largest sphere (such as the one shown in red on the cube picture) that could exist inside the pyramid without overlapping the other spheres?

Thank you Charlie.

The trick of mapping the vertices of a regular tetrahedron onto the vertices of a cube with center on the origin and sides parallel to the x, y, and z axes is one that I learned early in my math competition career, and one that has served me well in math and physics classes.