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Optimal result (Posted on 2018-06-04) Difficulty: 2 of 5
A stick of length L is broken into n equal parts.

What is the maximal product of their lengths?

No Solution Yet Submitted by Ady TZIDON    
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What Ady wants (spoiler?) | Comment 3 of 11 |
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
  Posted by Steve Herman on 2018-06-04 18:00:06

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