How many cups does it take to make a sphere?

One lovely afternoon after one of our Chalkdust meetings, we were sitting in the pub when, all of a sudden, we found ourselves surrounded by plastic cups (pint-sized… who knows why). So we started wondering: what could we do with so many cups? A cup is basically just a section of a cone, right? So if we put one right next to another, and then add a third, and a fourth and so on, making sure that the tips of that imaginary cone are as close as possible, then we should end up with a gigantic sphere, except perhaps for the holes between the cups. Or so we conjectured. Continue reading