Date: Feb 1, 2012 9:50 PM
Author: Eric Goodwin
Subject: Circles puzzle

OK.

Imagine a circle A. Regularly spaced on this circle's circumference
are the centres of a number of additional circles who just touch
(inside circle A) but don't overlap each other.

If you have two such circles, spaced 180 degrees apart, they'll touch
at the centre of circle A, so their radii will be equal to that of
circle A, and their total area will be twice that of circle A.

If you have 1 million such circles, they'll be just very tiny, looking
like a thicker line on the circumference of circle A, and their total
summed area will be less than that of circle A.

What is the lowest integer number of circles whose total area is equal
to or less than that of circle A?
What are two positive real numbers of circles whose total area is
equal to that of circle A? Can these numbers be expressed as anything
other than arbitrary decimals?

Does the puzzle extend to higher dimensions?

Eric