Infinite Square-Circle Inscription

If the side of the biggest square is $a$ , then the biggest circle has radius $\tfrac12a$ , and the side of the second-biggest square is $a/\sqrt 2$ .

The area of the biggest square is $a^2$ ; that of the biggest circle, $\tfrac14\pi a^2$ ; that of the second-biggest square is $\tfrac12a^2$ .

If we take the second-biggest square and all that is inside it, we have a square ring with area $a^2 - \tfrac12 a^2 = \tfrac12 a^2$ . The blue region in this ring has area $\tfrac14\pi a^2 - \tfrac12a^2$ . Therefore the blue fraction of this square ring is $\frac{\tfrac14\pi a^2 - \tfrac12a^2}{a^2 - \tfrac12a^2} = \frac{\tfrac12\pi - 1}{2 - 1} = \tfrac12\pi - 1 \approx 57\%,$ which is greater than $50\%$ . Therefore the outer blue region is slightly bigger than the outer red region; the same holds true for all blue and red regions, because of similarity.

Thus, the blue region is greater than the red region.