There is a square and a circle. The two are scaled appropriately and translated so they touch at just three distinct locations.

They intersect at two adjacent corners on one edge of the square, and at the midpoint of the opposite edge.

The question is, what is the ratio of the circumference of the circle to the perimeter of the square?

What is the ratio of the perimeter of the circle to that of the square?

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Solution

There are multiple ways to solve this puzzle using geometery, alegebra, or trigonometry, but I found the easiest way is to use the intersecting chords theorem.

The intersecting chords theorem states that if two chords intersect in a circle, then product of their segment lengths is equal.

For this puzzle we will use the following chords.

We'll define the length of the side of the square to be 2a, and the diameter of the circle to be d

Applying the theorem we can derive an equation for the relationship between d and a.

Now all we need to do to get the solution is calculate the ration of the perimeter of the circle (circumference), to that of the square.

The answer comes out at 1 : 16/5π

What I like about this puzzle is that the solution is amazingly close to 1:1. If you were trying to calculate this using a manual (non-analytic) method you could be forgiven for assuming that unity is the correct answer.