How many cups does it take to make a sphere?

The Chalkdust team set you a challenge, for which the prize is a £25 Amazon voucher…


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.


But does it even make an actual sphere? Would it fit in the room we were sitting in? Were there enough cups in the bar (or even London) to make that sphere? Calculating the actual number of paper cups required (or an estimate at least) shouldn’t be too complicated, so we gathered all the necessary measurements and everyone came up with their own estimate. Only, things are not that easy: assumptions might be quite different and so we ended up with such a wide range of “precise estimates” that we decided to make it into a contest!

For the past weeks, we have been collecting recycled paper cups (we don’t want this blog to kill the rainforest!) and now we are ready to set up the contest.  The measurements of the paper cups that we use are the following:


The steps we will follow to make the gigantic sphere are quite simple:



  1. First, we remove the rims of the paper cups, so that adjacent cups touch each other.
  2. Then, we put a couple of them together and we fasten them with a paper clip, ensuring that the bases of the paper cups align.
  3. We construct radially outwards from the initial cup until we get a gigantic sphere!


Here is a picture, so that you can have an idea of what the sphere will look like (and we give you the hint that we will need quite a few more than the seven paper cups you see in the picture!).


There are many ways to estimate the number of cups required, just as there are many things you might want to take into account. Are the paper cups rigid enough? Are the distances between the bases exactly equal to zero? Do the imaginary cones actually meet at a point in the centre of the sphere? Will the weight of the cups force the Sphere10shape to be more elliptic, changing the number of cups needed? These are indeed many questions that one would like to know the answers to before trying to guess the number… but let’s be honest, not even we know the answers until we build the sphere! In the meantime, we are having fun constructing a gigantic sphere made out of recycled paper cups (and then, if the size of the sphere and the weather allows it, we may even play football with it!).

Rules for the contest

  • The rules are quite simple. You just have to submit your guess for the number of paper cups required to make a sphere in the form below.
  • One randomly selected correct answer will win a £25 voucher from Amazon.
  • Only one entry per person will be accepted.
  • Answers from every country are allowed.
  • Winners will be notified by email and announced on our blog by 23 June 2016.

This contest has now ended!

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