The two pictures together show how the twelve pentagons on the soccer ball are coloured: there are two rings of pentagons each surrounding a "pole" of the opposite colour. A symmetry is made of rotations and/or reflections which leave the (uncoloured) structure exactly as it was. You could, for instance, rotate the ball by 72 degrees around the "polar" axis -- leaving the colouring intact. Or you could reflect it in the "equatorial" plane and then rotate by 180 degrees -- swapping the colours.

How many symmetries of the ball do swap the colours? Further down on this page, you'll get a chance to submit your answer. To study the question, it might help to look at the icosahedron depicted at the lower left of this page. It is obtained from the ball by contracting each of the pentagons to a point, but it has the same symmetries -- 60 in all, some of which neither preserve nor interchange the colours.

Symmetries abound in nature and are very helpful in applied mathematics: equations governing a physical system inherit its symmetries and thereby often become tractable. Symmetries play a leading role in quantum chemistry and solid state physics -- for instance, in the study of recently found super-conductors. They are also important in coding theory (for digital messages) and such mathematical triumphs as the successful assault on Fermat's Last Theorem.

Icosahedral symmetries -- though still on the leading edge of research -- apply to the quantum theory of the "bucky-ball", an amazing carbon molecule with sixty atoms (depicted on the right) which occurs freely in the smoke of candles. Named after Buckminster Fuller, it is the ancestor of a huge family of molecules, called fullerenes, which promise to be the elements of tomorrow's nano-technology

The Pacific Institute for the Mathematical Sciences (PIMS) is a non-profit organisation supported by five universities of Western Canada and dedicated to the promotion of mathematical research -- but it also has a programme of education and public awareness.

This contest is now over. The correct answer was 10 -- the winner: Katy Cheng.