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### Number and algebra

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# Partly Painted Cube

How many different ways could you paint two faces of a cube? Or three? Or four?

Can you work out a way of counting how many cubes are left unpainted in each case?

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Age 14 to 16

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How many different ways could you paint two faces of a cube? Or three? Or four?

Can you work out a way of counting how many cubes are left unpainted in each case?

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.