Banach-Tarski paradox: minimum number of pieces to get from a sphere of any size to a sphere of any other size? I just read about this amazing result and I have a very short question. I've read that five pieces is enough to get from one sphere to two spheres: is the underlying process sort of the same to get from one sphere to a sphere of any size (is five enough?)? Sorry if this has been asked before, I couldn't find anything. 
 A: To begin with, see here. 
I do not know of a written account where details are worked out more precisely, but the basic idea is the following: Lebesgue outer measure on $\mathbb R^n$, $\mu^*$, is defined for all subsets of $\mathbb R^n$, whether measurable or not, extends Lebesgue measure $\mu$, and satisfies monotonicity: $\mu^*(A)\le\mu^*(B)$ for $A\subseteq B$, and subadditivity: $\mu^*(\bigcup_{n\in\mathbb N}A_n)\le\sum_n\mu^*(A_n)$.
Suppose a sphere of volume $V$ is split into $k$ pieces $A_i$. Then $\mu^*(A_i)\le V$ for all $i$, so $\sum_i\mu^*(A_i)\le kV$, that is, no more than a sphere of volume at most $kV$ (or $k$ spheres of the same radius as the original) can be obtained if only $k$ pieces are used. In the link given above, Blackadar mentions that, in his estimation, about $10^{30}$ pieces are needed to turn a pea-sized ball into one of the size of the sun. 
Barry Cipra pointed out in the comments that the outer measure argument is an overkill for the pea-sun analysis: If a sphere $S$ has diameter larger than $kd$, where $d$ is the diameter of the original sphere $s$, then along any diameter it has at least $k+1$ points at distance larger than $d$ from each other so, if $S$ is obtained by splitting and rearranging $s$, at least $k+1$ pieces are needed. 
We can say a bit more: For $n>1$, let $f(n)$ be the smallest $k$ such that a solid sphere can be split into $k$ pieces that can be rearranged to form $n$ solid spheres of the same radius as the original. Each sphere must use at least $2$ pieces, so $f(n)\ge 2n$, and we know that $f(2)=5$. It would be tempting to say that $f(n)\le 2n+1$: For example, start with a sphere, split into $5$ pieces, and rearrange, so we have pieces $A,B,C,D,E$, with $A,B$ forming a sphere and $C,D,E$ another. (It would be natural to expect that) we can split $E$ into three pieces, isomorphic to $A,B,E$, and further rearrange; but this is a bit careless: Even if $E$ is chosen of size continuum, the pieces $C,D$ must be moved in this process, and end up being split as well. The best I can see at the moment is $f(n)\le 5n-2$ for $n>2$.
