Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?

Take any pair of two digit numbers x=ab and y=cd where, without
loss of generality, ab > cd . Form two 4 digit numbers r=abcd
and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$
for different choices of the first two terms. Make a conjecture
about the behaviour of these sequences. Can you prove your
conjecture?

Curvy Areas

Stage: 4 Challenge Level:

Suppose the radius of the smallest semicircle is $x$.
What are the radii of the other semicircles?
What areas can you work out, in terms of $x$ and $\pi$?