Monday, January 6, 2014

Counting to 41,044,208,702,632,496,804 and beyond...

Counting is a rather simple thing. You increment a given number by one just like in the journey of one thousand miles you take a step and repeat. Trouble is knowing when to stop. Ultra runners might not stop until they get to 100 miles (about 180,00 steps), but few will be able to tell you exactly how many steps they took. It's just too annoying to keep the count up and enjoy the view. Regardless, this gives us a good estimate of how high a person might be able to count in a day (if they count really fast). To get to the number in the title, it would take such a person over 600 billion years...

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.

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