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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.

Square Number Surprises

Age 14 to 16
Challenge Level

Why do this problem?

This problem offers students an opportunity to explore particular numerical cases which give rise to patterns that they may be keen to explain. When they are ready, students can make generalisations, and appreciate the power of algebra to capture the generality in a concise and elegant way.

Possible approach

This printable worksheet may be useful: Square Number Surprises.

You may wish to split the class into groups and give one of the four challenges to each group. Alternatively, you could introduce the first challenge and work through it as a whole class, and then set students the remaining three challenges to try. 

Encourage students to start with lots of numerical examples first, and then to write down everything they notice about their answers. Then share ideas for how to explain what they have noticed. If nobody suggests doing so, draw attention to the idea of using $n$ to represent a general number, and encourage students to write expressions for the general term for each of the four challenges.

The first two challenges lead to an expression which includes the term $n(n+1)$ so there's a good opportunity to discuss how we know that $n(n+1)$ must always be even. Students may make the connection with the formula for the triangular numbers $\frac{n(n+1)}{2}$, which always gives a whole number.

The second pair of challenges could lead to the expansion of $(n-1)(n+1)$ to give $n^2-1$ so this could link to work on the difference of two squares - see Hollow Squares and What's Possible?.

Key questions

Which numbers might it be a good idea to start with?
When we try some larger numbers, do the patterns we've noticed still hold?
How could we prove that the patterns will continue?

Possible support

Quadratic Patterns offers an exploration of quadratic number patterns using multiple representations, so might be a good task to ease students into this sort of generalisation. 
Pair Products offers an accessible and scaffolded context for exploring expansion of double brackets.

Possible extension

Students might want to try Always Perfect and Perfectly Square next.