Consider the smallest ball - clearly this ball can not touch $33$ balls of its own size or greater.

laegolas wrote: Consider the smallest ball - clearly this ball can not touch $33$ balls of its own size or greater. Why though?

Still.. Why though? What's a complete proof?

Here’s a way to find a complete proof (can’t type up full thing because I’m at school): Find maximum number of spheres (x) that can be externally tangent to a central sphere (all of the same radius- r). Do this by drawing segments connecting centers, etc. After this, it can be seen that if only x spheres of equal size can be tangent to a central sphere, than less than x spheres of greater size can be tangent to a central sphere. This is the situation of the smallest sphere in your 100-ball scenario, as described by Laegolas above. If x<33, then you’re done. Does that work?