Suppose we are given a set of circles with integer radii 1, 2, 3 ... $n$. What is the smallest square which they can all fit in such that they do not overlap?
For instance, when $n$=1, clearly we can fit the circle of radius 1 into a square of side length 2 ($s$ = 2). The circle fills $\frac{\pi}{4} \approx 0.785 $ of the area of the square.
When $n$=2, it is obvious that the circles should touch each other, and that $s=3+\frac{3}{\sqrt2}$. Here, the circles fill $\frac{5\pi}{13.5+9\sqrt2} \approx 0.599$ of the area of the square.
When $n$ = 3, we once again place the largest circles touching each other, and place the square around them. There is plenty of space for the circle of radius 1. $s=5+\frac{5}{\sqrt2}$. Here, the circles fill $\frac{14\pi}{37.5+25\sqrt2} \approx 0.604$ of the area of the square.
When $n$=4, once again the 4- and 3- radii circles determine the minimum size of square, and there is space for the 2- and 1- radii circles to fit in around. This means the circles fill $\frac{30\pi}{63.5+49\sqrt2} \approx 0.66$ of the area of the square.
However, when $n$=5, the 3-radius circle can no longer fit within the square determined by the 5- and 4-radii circles. The situation becomes considerably more complex as $n$ increases.
So my questions are:
Is there an algorithm (or approximation) which optimally arranges the $n$ circles in a square?
What does the area of the circles, as a proportion of the area of the square, limit towards as $n$ increases.