You are an architect designing a desert palace. One of your rooms is circular, having a diameter of 6 meters. You want to fill as much of the circle as possible with a symmetric design of square tiles of various sizes. You have to decide the size and placement of the squares.

You want to create the design starting with a single large square in the center having side length x. Assuming that you want to have more than one square in your design, the pattern for adding squares continues as follows: four squares of side length x/2, then 12 of side length x/4, then 36 of side length x/8, and so on. That is, after x/2, keep increasing the numbers of squares by a factor of 3 and decreasing the side length by a factor of 2, until the next size would cause squares to overlap. Note that the total area of the next squares of side length y (for y <= x/2) is 3/4 of the area of the squares of side length 2y.

Problems

2. What should x be to maximize the area covered by the tiles, keep the squares inside the circle, and keep them from overlapping, assuming a design like the one here?

In this design, a radial line goes through each side of the central square and, if you choose to use more squares, intersects one square of each smaller size. Further, each square has smaller squares in the middle of all its available sides. How much of the circle would then be covered?

Suppose that the constraints on the design are reduced to the following symmetry conditions: rotating the figure by any multiple of 90¿ should leave the enclosed squares looking identical. Furthermore, there should be lines of reflection about the middle of the central square vertically and horizontally and through the two diagonals. That is, the two sides of each line of reflection should be mirror images of one another.

2. How much area could be captured in a symmetrical design of this sort?

A solution will appear on October 21, 2004. In the meantime, have fun competing with your friends.