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

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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:
Well done to Jim sent us his solution:

The trick to solving this problem is to work systemticlly. The 8 weight can be placed on either the end or the inner point, and it is symetrical so we only have to look at on side.

If the 8 is on the end the arrangements to consider are:

If the 8 is on an inner attachment point we need to consider:

I them tried to find the balance points, and found they were in the central interval except for 8241, 8412 and 8412. I found that 4821 and 8142 produce a balance point at one of the inner attachment points.

For 8241, and 8412 the bar needs weight 4/3, for 8421, 8/3 is required. So the bar must be at least 8/3 in weight for all arrangments to have balance points in the central interval.