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Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

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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}.

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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.

Reasonable Algebra

Age 14 to 16 Challenge Level:

This problem invites students to explore the power of reasoning based on algebraic forms.

It is important for students to feel confident with the context and therefore spending time on the problem "Consecutive sums" may be an important starting point. By working on this problem the difficulties with solving for powers of 2 often occurs naturally.

Why are powers of 2 impossible? We are aware of two approaches to explaining this:
  • one is using an argument based on the diagram offered in the hints and the relationship of the rectangle with odd and even numbers.
  • the other involves examination of an algebaric expression that represents the sum - this links with the arguments offered in the solution to the problem "Make 24".
But of course there might be some neater explanations.

The problem Sequences and Series has an interactivity you might wish to use.