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 \[2\pi a^2 h\]
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