> 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
let A be a unit circle, then the area Sn of n circles placed on A as described is:
Sn = n*pi*(sin(pi/n))^2 .
The area of A is:
So = pi .
For n = 10, Sn < So .
The puzzle can not be extended to higher dimensions. For example in 3D there are only 5 regular polyhedra, in contrast to 2D where there are regular polygons of any number of sides.