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First note that there are 10¹⁰ possible combinations of the numbers from 1 to 10: there are 10 choices for the number that goes in the first position, 10 choices for the number that goes in the second position and so on. Since each digit is equally likely, each of these combinations is equally likely, so the chance of picking an individual combination is 1/10¹⁰.

Imagine picking the four hymn numbers out of a hat. First note that four-hymn combinations with one 1-digit number and three 3-digit numbers come in four types: the one digit number can occur in first, second, third or last place of the selection. So the overall chance of picking such a combination is equal to:

What's the risk of passive smoking? Or climate change? How big is the terrorist threat? And should we trust league tables?

If you've ever redecorated a bathroom, you'll know that there are only so many ways in which you can tile a flat plane.

A brief introduction to complex numbers

Complex numbers are based on the number i which is defined to be the square root of -1, so i times i equals -1. This number isn't a real number, in other words it does not appear on the usual number line. For this reason it is called an imaginary number, a slightly contentious name. Now any complex number is of the form a + ib, where a and b are ordinary real numbers.

No simple polyhedron has seven edges.

Proof: We show first that for any polyhedron we have 2E ≥ 3F and 2E ≥ 3V. The faces of the polyhedron are polygons, each bounded by a number of sides. Along each edge exactly two faces come together, so an edge corresponds to exactly two sides: the total number of sides is 2E. We also notice that any face has at least 3 sides, so the total number of sides is at least 3 times the number of faces.

The following diagrams show our algorithm as applied to the network arising from the dodecahedron.

Computer generated movies and electronic games: Joan Lasenby tells us about the mathematics and engineering behind them.

The red lines show the three perpendiculars from the vertices to the opposite sides meeting in the orthocentre O, the blue lines show the three lines from the vertices to the midpoints of the opposite sides meeting in the centroid G and the green circle surrounds the triangle and the circumcentre P. These three points lie on a line with the centroid one-third of the distance between the other two points.