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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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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.

Attractive Tablecloths

Age 14 to 16 Challenge Level:


How many extra colours do you need to add to the $7$ by $7$ pattern to make the $9$ by $9$ pattern for each type of symmetry?

Where do those extra colours go?

How many extra colours would you need for the $11$ by $11$ design?

Where do those extra colours go?