I came up with an interesting question for myself, but I can't find the answer. Here are the rules:
You take any number, let's say we take 9.
You're then allowed to use any group of positive numbers whose sum is 9, and try to get the biggest product out of them.
For example, you might try 4*5 or 3*3*3 or 4.5*4.5, etc. There is no limit to how many multiplications you use.
That's pretty much it. Basically I wanted to see if there was some sort of formula for always finding the greatest product for any number while abiding by these rules, or any way of showing what the greatest product was even for a particular number.
I came up with this: (N^0.5)^(N^0.5) which as far as I know abides by the rules above - although I don't know how to write it out with normal multiplications - but it turns out not to be the answer after all. Any help or comments much appreciated!