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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.

Triangles Within Pentagons

Age 14 to 16 Challenge Level:

Having identified the rule is it possible to work backwards and reconstruct pentagons using the triangular numbers?

This problem builds on the work of the three problems "sequences and series ", "triangles within triangles " and "triangles within squares ".

There are many more patterns and relationships involving triangular numbers and some experimentation could result in some pupil generated discoveries.