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