Is this a good recreational math problem?

I thought of this problem, though I am sure I am not the first.

Pick some number, say 12. Look at groups of positive integers that add to 12, for example 5, 5 and 2. Multiply these numbers, 5×5 x 2= 50. We want to find the group that gives the largest value when multiplied.

At first the problem seemed really difficult. Then with a little trial and error, I began to intuit what was going on, and found what seemed to be the general solution to all problems of this type. I think most others could do the same within a few minutes. I don’t think that you can immediately guess what the answer is. It came as a bit of a surprise.

The only downside is that to actually prove that the solution is correct requires a very elementary application of logarithms and calculus. The proof is of interest in itself, and I encourage anyone with the right background to give it a try. I will outline the proof later.