Did you solve it? The Pi day puzzle that will spin you in circles

The solution to todays topical teaser

Earlier today I set you the following puzzle:

Illustrated below is a quarter-circle, containing two semicircles of smaller circles. Prove that the red segment has the same area as the blue.

Explain why red = blue


Place four identical quarter circles together to see the full circle. We can see that it comprises four smaller circles that overlap.

The radius r of the large circle is marked

From looking at the areas in the diagram we can deduce the following:

  1. The area of the four overlapping smaller circles (the white and blue sections) is equal to the area of four of these circles minus the area of the overlap (the blue section).
  2. The area of the whole circle minus the area of the red section is equal to the area of the white and the blue sections.

Its Pi Day! I told you that the area of a circle is pi times the radius squared. So the area of the large circle is r2.

The radius of each of the smaller circles is half r, orr/2. So the area of each smaller circle is (r/2)2 , or r2/22, which isr2/4.

So, the area of four smaller circles is 4 x r2/4 = r2.

We can now rephrase the two statements above as follows:

  1. The white and blue section is equal to r2 minus the blue section.
  2. The white and blue section is equal to r2 minus the red section.

If these statements are both true, the blue and red sections must be equal. Since each quarter has the same ratio of blue to red, each blue section must have the same area as each red.

This puzzle appears in The Tokyo Puzzles by Kobon Fujimura.

I post a puzzle here every second Monday. My most recent book is the mathematical adult colouring book Snowflake Seashell Star. (In the US its title is Patterns of the Universe.)

You can check me out on Twitter, Facebook, Google+ and my personal website. And if know of any great puzzles that you would like me to set here, get in touch.

Read more: https://www.theguardian.com/science/2016/mar/14/did-you-solve-it-the-pi-day-puzzle-that-will-spin-you-in-circles