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Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.


The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

Lap Times

Age 14 to 16
Challenge Level

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds. If you record the times each takes to complete a lap can you devise a method for finding when and where they will meet? You may like to start by supposing they take 72 seconds and 80 seconds respectively to complete a lap. Notice that with these lap times they meet for the first time at the starting line. When and where do they meet if the lap times are 70 seconds and 85 seconds? Can you find other lap times that are such that the cyclists meet exactly half way round the track.