Rearrangement lemma. "It's just common sense"

If I have loads of cars, loads of cds, and loads of chocolate sprinkles, and you get to take 3 of one thing, 2 of another, and 1 of the third, what would you do?

3 cars, 2 cds, 1 chocolate sprinkle.

That's because to maximise the sum, you need to pair the largest items together (3 and car) and the smallest items together (1 and chocolate sprinkle).

Any other mixture, say, 2 cars, 3 chocolate sprinkles, 1 cd, will never be as big.

A and B are interchangable in the above formula, so we'll say A>B, and so A^2>B^2 cos they're whole and unequal.

We're matching up A^2 and B^2 with A and B.

Match biggest with biggest: A^2 with A

Match smallest with smallest: B^2 with B

and we get A^3+B^3 being bigger than any possible other mixture. In the equation, the mixture matches A^2 with B and B^2 with A.