While trying to come up with an interesting math problem I came up with a solution to the following:
Consider $N$ equally spaced points on a circle of radius $1$. Line segments connect these points such that each point is connected to every other point by exactly one line segment. What is the product of the lengths of all of these line segments?
For clarification, the resulting graph (a common doodle) for $N$ = $13$ looks like this:
The solution I found is that the product $P$ is given by
$P=N^{N/2}$
I'm sure this same problem has been solved before and with more rigor but I have not been able to find a solution to this problem (or even the problem statement) online. Does anyone know of a proof for this solution or know where to look to see if someone else has solved this problem?
Use complex numbers. The points can be regarded as $$e^{2k\pi i/N}\quad\hbox{for}\quad k=0,1,2,\ldots,N-1\ .$$ The product of all the lengths from $1$ (that is, from the point with $k=0$) is $$\prod_{k=1}^{N-1}|1-e^{2k\pi i/N}| =\left|\prod_{k=1}^{N-1}(1-e^{2k\pi i/N})\right|\ .$$ Now the product is $$\lim_{z\to1}\frac{z^N-1}{z-1}=\lim_{z\to1}\frac{Nz^{N-1}}{1}=N\ .$$ The product is the same for all $N$ roots so we get $N^N$; but every length has been included twice ("once from each end"), so the correct answer is $$\sqrt{N^N}=N^{N/2}\ .$$