How Did pi Get Here? (FWN’s Solution to 3Blue1Brown’s Puzzle)

Joining the discussion today.

Here’s the problem in its animated glory (From 3blue1brown on Youtube)

I guess that I have a reasonably readable overview of the problem in the solution below, but here is another attempt to summarise it.

If we set up a model in which two point masses sit on a frictionless, horizontal rod/table on the same side of a wall at the origin, and we push the heavier one on the outside inwards to set the system in motion.

Assuming elastic collision everywhere, the total number of collisions in this system is connected to our favourite circle constant, π.

Namely, if the heavy mass is 100^(n-1) times the small mass, we recover the first n digits of pi.

Unexpected?

The working above is most certainly high school level and it is exactly where I first encountered this problem.

There are obviously more approaches. One may as well develop a higher perspective in which we treat every block-block collision as a special O(2) linear operator, or one may even devise a billiard problem related to the phase space that we defined above. I did not write this (although billiards are literally my current research project) as I saw them in 3b1b’s video and am curious on his take.


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