Let’s play a game where you have ten chips, and you want to split them up into stacks such that you want to maximise your score, which is the product of the size of each stack. What’s the best you can do with 10? How about 100 chips? How about N chips arbitarily large?
I'm not sure how to go about this problem, but through brute force, I got 36 with 10 chips (334=36). Any ideas?
Let’s forget about the discreteness for a minute and make stacks of real-valued size $s$. Then the product is $s^{N/s}$. Maximizing the logarithm of the product yields
$$ \log\left(s^{N/s}\right)=\frac Ns\log s\to\max\;, $$
$$ \frac1{s^2}-\frac{\log s}{s^2}=0 $$
and thus $\log s=1$, i.e. $s=\mathrm e\approx2.7$. So the stack sizes should be $2$, $3$ or $4$, with $2$ stacks of $2$ being equivalent to one stack of $4$. Two stacks of $3$ use as many chips as three stacks of $2$ but yield a higher product, so most of the stacks should be $3$ and only what’s left should be in a stack of $2$ or $4$. For $N$ chips this yields a product of
$$2^{N\bmod 3}3^{\left\lfloor\frac N3\right\rfloor}\approx\sqrt[3]3^N\approx1.44^N\;.$$