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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.

Balance Point

Age 14 to 16 Challenge Level:

For this problem you need to be familiar with the principle of moments.

Have a look at a problem called "Inside Outside" to explore this idea.

Then come and try this again.

Back with Balance Point :

How many possible arrangements were there to choose from?

How can you list these systematically, and efficiently eliminate anything that is already covered by another arrangement?