New” Objects in Problems

This 3b1b video points out a nice heuristic for how to find solutions to a general problem.

The “source” problem there is about randomly choosing points on a sphere and asking what the probability is that the induced tetrahedron contains the center of the sphere.

But the “algorithm” to find a solution is nice:

  • Simplify the problem to a simpler environment (2D circles and triangles in this case)
  • In that environment, notice that a new object simplifies the way to view the problem (in this case, noticing that instead of considering 3 random points, consider 2 random lines being drawn through the circle, and then an assignment of 2 base points from the 4 possible intersection points, and therefore to the answer of 1/4 probability)
  • Regeneralize to the initial environment

In short: simplify, then notice whether a new object is used, then recast the problem in terms of the new object, then re-generalize the recasted problem.

Also — combinatorics is great.

Category: math