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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 Triangles

Age 14 to 16
Challenge Level
It is usefult to be able to recreate these patterns on square dotty paper.
Is the way the triangles fit together always going to work with two consecutive triangular numbers?
Why?

Yuu might find it useful to look at the problem on "Sequences and Series".