### Problem

Given a 1×1 square. Is it possible to put into it not intersecting circles so that the sum of their radii will be 1996?

### Solution

It's an interesting problem. The number 1996 looks pretty much random and irrelevant. Chances are that either the problem in solvable for 1996 or it is solvable for other and larger numbers as well. Let us experiment. Start with simple configurations. If there is just one circle, how big its radius could be? 1/2. If my purpose at this point is to gain an insight into the problem, what simple configurations can I use? 2 circles? The result will be a half of the square's diagonal - .

So I'll skip the case of three circles. Take four. Here's one possible configuration. The sum of radii is 1. In three experiments we obtained an increasing sequence of numbers: 1/2, , and 1. It may pay to continue with our experiment using simple configurations. Looking at the two pictures, it was easy to find the radius of a circle when it was inscribed into a square. First, it was the original 1×1 square. The last time, we had 4 smaller squares of the size 1/2x1/2 each. Dividing the square into nine 1/3×1/3 squares we'll get the sum of radii 1/6·9 = 3/2 - a new increase. With 16 squares the sum will be 1/8*16=2. With n2 small squares, the sum will become 1/(2n)·n2 = n/2.

Since n can be chosen arbitrary large there is no limit to the sum of radii of the circles packed into a 1×1 square.