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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}.

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.

Chocolate 2010

Age 14 to 16 Challenge Level:

This looks like a fairly standard problem leading to a simple algebraic equation that can be solved, however the need to examine what is happening to each of the digits of the number means that the solution requires a little more thought and an appreciation of place value.

Being able to generalise this to any year may help convince you that the structure of the problem is understood.