Math is Patterns

If you’re walking 2 laps while a friend is running 3 laps around the same track and your speeds are synced up just right so you both begin and end at the same time, then at what point in your set does your friend pass you? What if you’re doing 4 laps and your friend is doing 5? Or 2 and 4?

I was surprised by the answer: your friend will not actually pass you. Despite doing more work, they’ll catch you at precisely two points, the beginning and the end. This applies any time your friend is doing one more lap than you. In the general case where you do m laps and your friend does n (w.l.o.g. n strictly greater than m) you will be passed (n-m)-1 times.

One way of seeing this is to fix your frame of reference to your position by imagining that you do not move but rather that the track is revolving around you. Your friend is running their 3 laps but the track is moving them back 2. So relative to you they are only running 1 lap while you stand still at the start location. This is the sort of model that makes the general case intuitive.

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