Now consider how long it would take to count the number of possible distinct non-recrossing paths following the edges of a square between the opposing corners along a diagonal. That's two, and it takes you less than a second. Suppose that you make it a two by two square. The number of possible distinct non-recrossing paths now comes up to 12. You might then ask about a 3 by 3 square or larger. Or maybe not as that might seem a little too mathematical. Dr. Minato and his colleagues followed this train of thought and made a YouTube video to illustrate how quickly the count grows. It's in Japanese with English subtitles and already has well over a million page views!
While this is a very fundamental question, it's useful to recognize that knowing how to count paths (particularly using Minato's clever algorithms) on an arbitrary network has lots of cool applications. Among these could be the determination of the sum of chemical pathways between reactants and products. That's the problem that I'm interested in, but it's a little harder because each path has a different weight (or cost.) The cost isn't necessarily the same for each of Minato's subsets and consequently it isn't trivial to reuse his existing algorithms. But here lies a challenge to a possible advance in the field of chemical physics.
Check out "The Art of 10^64 -Understanding Vastness-Time with class! Let's count!" on YouTube.