breaking sticks of various length

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A riddle from the Riddler with a variation on the theme of breaking sticks: Given a stick of length L, what is the optimal manner to break said stick to achieve a maximal product of the individual lengths? While the pen & paper resolution is a one-line back-of-the-envelope calculation, with an impact of the length L, obviously,  a quick R code leads to an approximate solution:

mw=function(k=2,l=10,T=1e6){ a=matrix(runif(T*k),k) for(i in 1:T)F=max(F,prod(l*a[,i]/sum(a[,i]))) return(F)}

with increasing inaccuracy when L grows, obviously.

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