Today Professor Pheno Menon challenged me with a puzzle:
"I will give you a composite number $n$. You need to give me a list of $n$ integers, all between $1$ to $n$ (those two included), so that the sum of their cubes is equal to the square of their sum. Can you always do that?
I said "That's easy! I can just give the numbers $1,2,\cdots,n$."
He replied, "Not so fast, my friend! There are some more conditions: the list must have a unique minimum and a unique maximum, and it mustn't be exactly the same as $1,2,\cdots ,n$ (up to permutation, of course). Now tell me, is it always possible?"
I tried for some time, but couldn't quite get to any conclusion. Maybe you can help?
I'd prefer a solution with completely elementary maths, not higher than high school mathematics.