Product of All Chord Lengths on a Unit Circle: Why the Answer Is n^(n/2)

A Daily Math Challenge for the Weekend

So that the weekend isn’t spent thinking only about jokes and the role of mathematics in the fate of humanity — here is a small problem to chew on.

Unit Circle Chord Product: The Problem

Imagine n > 2 points placed evenly around a unit circle. Draw a line segment between every pair of distinct points. The question: what is the product of the lengths of all such segments?

Hint: Complex Numbers

Think about complex numbers.

The Answer to the Unit Circle Chord Product

The result is surprisingly clean:

n^(n/2)