A stick of length L is broken into n equal parts.
What is the maximal product of their lengths?
If n is a variable, then a little experimentation with different L and excel seems to indicate that the product is maximized when n = L/e, where e is the natural base of logarithms. Then the product (L/n)^n would be equal to e^(L/e).
Of course n is an integer, so it is necessary to take both the ceiling and the floor of L/e, and see which leads to the maximal product. That product will necessarily be less than e^(L/e). In most cases one can just round (L/e) to the nearest integer, but that does not work 100%. In particular, if L = 3, then the maximal product occurs when n = 2, even though (L/e) rounds to 1.
In other words, maximal product =
max((L/ceil(L/e))^ceil(L/e) , (L/floor(L/e))^floor(L/e)).
Sorry, I do not have a proof available,
Edited on June 4, 2018, 6:01 pm