# Dynamic Systems

What does doing laundry have to do with math? Well, as it turns out, just about everything! As a mathematician, I tend to be not only one who notices patterns but also one who seeks to identify and solve problems. I know, many people think mathematicians just go around solving problems. Actually, we go around thinking about how mathematically cool everything is and occasionally get in fender benders not because we are texting and driving, but because we are in a math haze. Yes, we also identify and solve problems. One of my biggest “problems” every summer is doing laundry. We live in a cabin with no washer and dryer so about once a week I haul our dirty clothes / towels / sheets to a laundromat. Can math solve this problem?

For starters, we are in New Hampshire for about 49 days. So if I do laundry every seven days, I’ll have to do it seven times. But if I extend by just one day and do it every 8 days, which does not create too much more laundry volume, I’ll only have to do laundry 6 times. This means instead of spending only 14 hours of my summer doing laundry, I will spend only 12. Of course I could do laundry even fewer times, but that would require us to bring a lot more clothes / sheets / towels, and the likelihood of getting enough machines to do it all at once would diminish considerably.

So I have adopted this “once every 8 days on average” timing. Yesterday was Wednesday. I did laundry. I did it on a Tuesday the week prior and on a Monday the week before that. So next week I will probably do laundry on a Thursday… Maybe. The more you think about a problem, the more you realize there are many variables, and of course part of being a good problem solver is to identify the variables. Using this add-a-day method bumps me into the weekend. More about that later.

The local laundromat “Speedy Wash n Go” is a paradox of its own due to the dynamic system under which it operates.  If you arrive at a time when all of the machines are full, then you have added a minimum of 30 minutes to your task, maybe more depending on how many machines you need and how many other people are waiting for machines.  In this case it isn’t so speedy. I, of course, am always hunting for the elusive time at which I’ll have the place to myself, however, I am not interested in doing our laundry at 3 AM, though I doubt I’d be alone in there anyway.  I usually try to get there by 7 AM and this has been working for me. Luckily, I am not teaching right now so I am not hemmed in by a time constraint.

The one thing about the Speedy Wash n Go which is predictable is its unpredictability.  If I do laundry on the 4th of July, will I be the only one there since everyone else is “vacationing” or will the place be jammed because everyone is on vacation and needs to wash all of their towels? Is it better to do laundry on a sunny day or a cloudy day or a rainy day? For example, on a sunny day will the laundromat be empty because everyone is at the beach or on a boat, or, will it be packed because lots of people line-dry their garments in the fresh air? How about the weekend? This may not be an optimal time as camp counselors from the 12+ camps in the surrounding area might have only this time to do laundry in between the changing of the sessions. Plus, folks who work M – F might do laundry on Sat / Sun. This is where calculus and chaos theory meet psychology. If that’s not a dynamical system then I don’t know what is.

So, I try to optimize my laundry start time and day to minimize my time there. Not only are the time and day important, but so are the number of machines I need (I try to never need more that 3 of those massive front loaders). So far this summer I have managed a weekday 7 AM arrival and have not had any issues getting three front loading machines. Did math solve my laundry problem? Probably not — but thinking mathematically about all of the variables and their interactions got me closer to the best case scenario. Not every problem has a solution; sometimes the best we can do is manage.

If you want to learn more about how mathematicians think, author Steven Strogatz has a remarkable ability to highlight the joy, beauty and usefulness of math in various seemingly non-math settings.

1. AJ