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Diophantine N-tuples

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

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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}.

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Sixational

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:

If the track length is L then you can write down the speeds and the distance between the cyclists at any given time in terms of L. The cyclists meet when the relative distance covered is a multiple of L. You might like to model this using a cardboard tube and threads. Think of the length of the tube as the time axis and the threads around the tube showing the cyclists path on the track.