For a chance to win the first prize (£100 Amazon voucher), or one of our prizes for runners-up, submit your typed or neatly written solutions of the following problems tomaths@lincoln.ac.ukor by post to Mathematics Challenge, School of Mathematics and Physics, University of Lincoln, Lincoln, LN6 7TS. Please include your full name, postal address and email, as well as the name and address of your school. The closing date is 5 January, 2016. The competition is open to all young pre-university people in UK aged 16–18 years. It is not open to current university students. See full Terms and Conditions.

1. Suppose that an underground tunnel of length 100 km is constructed strictly along a straight line between two points on the Earth’s surface. Assuming that the Earth is a perfect sphere of radius 6,371 km, estimate the maximum depth which the tunnel reaches below ground.

2. Find…