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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}.


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.

Lens Angle

Age 14 to 16 Challenge Level:

Why do this problem :

This superficially fixed problem is an excellent way to draw students into the discovery of a general result. There are likely to be a range of possible routes to solution giving ample opportunity for discussion of other people's approaches and the elegance and efficiency of approaches.

Possible approach :

It seems at first as though there's not enough information to fix the angle whose size we are asked to establish - parts of the construction seem to have a lot of freedom to wander.

But as this situation is explored more it becomes apparent that the angle of interest maintains its size wherever the unconstrained parts of the diagram happen to rest.

This freedom to wander may be clear to the group almost immediately but if it is not, ask the students to reproduce the figure using the given values. This should help them appreciate what hasn't been specified, and lead them to question whether it is necessary for these to be specified.

Now they have something to explore.

Dynamic Geometry may help with enquiry, but isn't essential. Two drawings of the figure with measurements of the angle should be enough to suggest a possible general result which can then be reasoned over.

In conjunction with the problem presented the following connected result can be included : allow one fixed length chord to move around a given circle, relative to a second chord of some other fixed length. Joining the end of each chord to the opposite end of the other will produce two diagonal lines which, it turns out, intersect at the same angle regardless of the relative position of the chords.

Key questions :

  • Can anyone see how we could know the size of the missing angle - or any ideas how we might try to find it ?
  • Any thoughts at all about the problem ?
  • Can you describe how to draw this diagram ? What do you do first ? Then what ?

Possible extension :

Trapezium Four

Possible support :

This is an opportunity for plenty of drawing and measuring of angles associated with circles. All the standard angles in circles results can be acquired through discover and the justification for each as it emerges will be an important exercise in geometric reasoning.