# 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?

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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