It was a rainy day and I was on the bus; not the most pleasant of experiences. I was staring out the side window, hoping that the rain would stop before I got to mine. And then, I noticed something mathematical. What could have been another dull and wet ride took a different turn.

The windshield wipers were moving at different frequencies. Most of the time, the wipers were out of sync. One would move and then the other would follow. But every so often the wipers found their groove and seemed to sync up for a wipe or two. For those of you unfamiliar with this phenomenon, I created a quick visual for you. Insert Desmos.

In-sync:

Out of sync:

I wanted to model the situation mathematically so I could understand what was going on. I pulled out my phone to time the wipers. The left wiper completed 10 strokes in 12.6 seconds. The right wiper made 10 strokes in 12.8s. This meant that the left wiper was operating at a rate of 1.26 strokes per second and the right wiper was operating at a rate of 1.28 strokes per second. Surprising that such a small difference can create such an interesting pattern eh?

Let’s drop the decimals and work with whole numbers; 126 and 128. Since we want the wipers to synchronize, we are looking for a time when both numbers are the ‘same’. This calls for, the least common multiple! After some number crunching, we find that the LCM of 126 and 128 is 8064. Converting back to a decimal, we expect the wipers to synchronize after 80.64 seconds.

This makes intuitive sense to me since most of the bus ride the wipers were out of sync. However, the above calculations tell us that every minute and 20 seconds, the wipers would sync up and go through a couple wipes in tandem.

I wonder why the wipers aren’t perfectly in sync to begin with? Was it a manufacturing defect? Do the wipers have independent motors? Perhaps perfectly in-sync wipers put bus drivers to sleep? Or perhaps the universe just wanted to give me a fun puzzle that rainy day.