# Sunday Times Teaser 2708 – Abracadabra

### by Nick Jones

Ali Baba melts down sixty gold beads to turn them into coins. The beads are spherical but have a central cylindrical hole cut through them whose length is one centimetre.   The coins have a thickness of half a centimetre and a diameter of two centimetres.

How many coins does he make?

One Comment

Unless we get into numerical methods, this is more about maths than it is about programming. Let the sphere have a radius of (r) with a cylindrical hole of half height (h) and radius (a).

The volume of the sphere is [frac{4}{3}pi r^3] The volume of the cylinder is [2pi a^2 h]

The volume of the two spherical caps (removed by the hole) is [see here] [frac{2}{3}(r-h)^2 (2r+h)]

and we also have (a^2+h^2 = r^2). Taking the last two volumes from the first and simplifying the result gives the volume of the remainder as [frac{4}{3}pi h^3]

which is counterintuitive at first since it doesn’t depend on the radius of the sphere.

If the coins are of radius (c) and thickness (d) a little more algebra quickly shows that [N_{coins} = frac{4h^3}{3c^2 d} N_{beads}] which yields an answer of 20 coins with the given values.

But lets do some Python anyway!

which gives