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.

(In reply to re: Possible solution by Steve Herman)

I assumed what was meant was that the circles 'covered' the rectangle in the sense of the densest possible packing i.e. there would still be some (minimal) gaps. This would make the rectangle bigger than the total area of the circles.

I have to admit I hadn't considered the alternative of 'completely covered', but on brief reflection I'm not sure it matters; in the first case there is a constant 'efficiency' factor, x, less than 1; in the second there is an equivalent 'inefficiency' factor, say y, greater than 1. This factor still cancels leaving 400pi.

To put it another way, in the first case the circles are inscribed within the corresponding hexagons, in the second case, the hexagons  are circumscribed by the circles.

Edited on January 31, 2012, 11:31 am

Posted by broll on 2012-01-31 11:02:56