Patrick (patrickwonders) wrote,

Making Fun Algebra Problems Funner

A month ago, a friend posted the following problem on Facebook. I just noticed it this week.

Drawing of a circle with radius r sitting on a line with two squares wedged against it.  One square has side-length 1, the other side-length 2.  There are six units between the squares.

The goal is to find the exact length of the radius r.

I love this kind of math problem. It has a bunch of features that make it a great, toy math problem.

  • It looks like it’s going to be easy, but at the same time seems at a glance to not have enough information
  • It looks like a geometry problem but only requires that you know:
    • All radii of a circle have the same length
    • A radius to a point where a tangent line touches the circle is perpendicular to that tangent line
  • It requires only very basic algebra:
    • Pythagorean theorem
    • Solving quadratics
  • The numbers in the problem are small, non-zero integers

I spent the next 25 minutes and six pieces of paper working the problem. About 20% of the time that I spent was rechecking my work. Why did I bother rechecking my work on a toy problem?

Warning: Spoilers ahead. If you like this kind of problem, stop reading now and play with it first.

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