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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.

Balance Point

Age 14 to 16 Challenge Level:

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Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. The bar strangely has no weight of its own.

Move the bar from side to side until you find a balance point. Is it possible to predict that position
(you might find it helps to look at the problem called "Inside Outside" first)

Which arrangements produce balance points outside the central interval (between the two inner attachment points for weights)?

If the bar now does have weight, what is the least weight it could have with no arrangements that produce balance points outside the central interval.

You only need the 1, 2, 4 and 8 unit weights for the problem set but the others are there to let you explore, conjecture and test - have fun.