3. In a square, the inscribed circle is drawn, in which an inscribed square is drawn with sides parallel to the sides of the original square. The area between these squares is colored dark gray. In the smaller square, the inscribed circle is drawn again with another inscribed square with parallel sides; the area between these is colored light gray. We repeat this process until there are 2023 dark gray and 2023 light gray areas. In the picture below, you can see the first steps drawn. The innermost square remains white. The total area of all dark gray areas together is exactly 1.

![](https://cdn.mathpix.com/cropped/2024_04_17_f4c54c3a9ff27948c305g-3.jpg?height=508&width=525&top_left_y=477&top_left_x=1322)

How large is the total area of all light gray areas together?
B3.

If we rotate the second largest square by 45 degrees, the vertices lie on the midpoints of the sides of the largest square, and we see that the largest square has twice the area of the second largest square. This situation repeats: the second largest square, in turn, has twice the area of the third largest square.

If the area of the third square is $A$, then the area of the second square is $2A$ and that of the first square is $4A$. Therefore, the outer light gray area is $2A - A = A$ and the outer dark gray area is $4A - 2A = 2A$. Thus, the outer dark gray area is twice as large as the outer light gray area. The next dark gray area is twice as large as the next light gray area, and so on. In total, the dark gray area is twice as large as the light gray area. Therefore, the light gray area has an area of $\frac{1}{2}$.