What's causing the bright white reflection a under the right hand ball?
It looks like it's being amplified. why isn't that happening on the left?
It's because of the material of the balls. The left ball is made of a specular material, so it reflects like a mirror. The right ball is made of glass so the bright white spot you're seeing on the ground is the light having refracted through the glass ball from the light source. This happens on bounce three because the light has to first hit the surface of the glass ball, travel through the inside of the glass ball and hit the other side, and then it can hit the ground on the third bounce.