For a new puzzle consider a circle of radius 1. Inscribed
in the circle are three smaller circles,
all of the same radius,
that touch each other and the sides of the larger circle. Hence
they are as
large as possible while remaining within the larger
circle. There is a small open gap between these
three circles in
the center of the larger (radius 1) circle. Find the radius of
the largest circle that will fit
into this gap. The same
question can be asked if four circles are inscribed in the radius
1 circle. In general,
suppose that n circles are inscribed in a
radius 1 circle. What is the radius of the largest circle that
can be fit
into the gap between them (in terms of n)? It is
worth pointing out that, although the problems for n=3 and n=4
can
be worked using ordinary geometry, the general problem for
arbitrary n will require trigonometry.