Intense Circle Packing
Suppose I have a square canvas of area , and several circles whose total area is less than .
Is is always possible to place these circles onto the canvas so that they are non-overlapping?
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1 solution
Let's say you have a square of side-length 100, and thus an area of A = 10000.
Case 1: You have two circles, one of radius 51 and one of radius 1. Total area = ( 5 1 2 + 1 2 ) * pi = 8174, well under 10,000, but the larger circle, with a diameter of 102, cannot possibly fit in a square of side length 100.
Case 2: You have 4 circles, all of radius 1. Total area = 4 * pi = about 13, less than 10,000, and if you draw a square of side length 20 in the center of the original square and put the 4 circles with centers at the corners of the interior square, they fit easily.
So, sometimes you can and sometimes you can't. It depends on the circles.