Some interesting shapes are mentioned in Derrick Niederman’s *Number Freak* (2010). Using identical matchsticks, what’s the smallest fully connected shape you can make in which two matches meet at every vertex? That is, what is the smallest 2-regular matchstick graph?

It’s an equilateral triangle:

Now, what is the smallest fully connected shape you can make in which three matches meet at every vertex? That is, what is the smallest 3-regular matchstick graph? It uses twelve identical matches and looks like this:

And here is the smallest known 4-regular matchstick graph, discovered by the German mathematician Heiko Harborth and using 104 identical matches:

But Niederman says that “it’s impossible to create any arrangement in which *five* or more matchsticks meet at every vertex” (entry for “104”, pg. 230 of the 2012 paperback).