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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.

A Tilted Square

Age 14 to 16
Challenge Level

There are several possible approaches to the final problem:

  1. Using Pythagoras theorem.
  2. Using the fact that the square can be split into 45 ° right angled triangles and some simple trig.
  3. Some symmetry and the differences in the x and y coordinates of the vertices and the coordinates of the centre of the square.