Prove, that if a given rectangle can be covered by 100 circles each of r=2, then it can
be covered by 400 circles each of r=1.
Let the 100 r=2 circles fill the rectangle with an efficiency x, which is a fraction less than 1. Since they are smaller, the r=1 circles must fill the same rectangle with at least the same efficiency of x. Now for the area of the rectangle we have 400pi/x, and the smaller circles fill that with an efficiency of 400pi/x*x, or 400pi.
So at least 400 small circles fit into the same area. Some testing with hex tilings suggests at least 408 small circles should be possible.
Edited on January 31, 2012, 5:26 am
Posted by broll
on 2012-01-31 02:01:07