Expanding Cubes

Firstly, note that the volume increases by a factor of 8 when the length doubles because $2^{3}=8$ . Hence, the volume occupied by the one cube over the ten days follows the geometric sequence $8, 64, 512, 4096...$ up to $8^{10}$ . However, if you start with 64 $2\times 2$ cubes then the volume occupied by cubes on the first day is $64\times 8=512$ , so you miss out the first 2 terms in the sequence. This means that it only takes $10-2=8$ days, so the solution is $\boxed{8}$ .