An Average Question: Upper Secondary Mathematics Competition Question

Calculate the average value of the expression

(a - b)² + (b – c)² + (c – d)² + (d – e)² + (e – f)²

over all possible arrangements of (a, b, c, d, e, f) of the six numbers 2, 3, 5, 12, 15, 16.


On the face of it, this seems like a long hard slog of enumerating 6! = 720 different arrangements of the six numbers. However, there is a shortcut!

This question has been adapted from a UKMT Cayley paper. This means it is aimed at the best lower secondary mathematicians in the UK – only the top 500 are invited to participate – so I think it should be solvable by upper secondary students. I’d rate this at about 10-15 minutes of writing and thinking time.




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