Isaac Newton and David Gregory argued about it in 1694. Newton said the answer was 12, Gregory thought 13 might be possible. In fact you can rearrange the 12 spheres in any way you like by sliding them without intersecting, always touching the central sphere. I mean, like you suggested, Chief. Seems that the Olympus Expedition found ancient Martian remains, including some very strange machinery.
Martian science seems to have been a bit weird.
What they didn't realise is that the machinery was still working. Everything suddenly looked different, but exactly the same. It was very disconcerting. They were right. Cosgrove shook his head sadly.