Sunday Times Teaser 2572 – Abominable

by Angela Newing

The diagram above shows half the layout involved in this teaser. The radii of the circles involved are $$r$$, $$3r$$, $$4r$$ and $$s$$. Since the angle $$alpha$$ is common to triangles ABD and ACD, we can apply the cosine rule for this angle in both triangles to give: $(r+s)^2+9r^2-6r(r+s)cosalpha = (4r-s)^2$ $(r+s)^2+16r^2-8r(r+s)cosalpha=(3r+s)^2$ we can now eliminate the (cosalpha) term and simplify the result to give: $4r(13s-12r)=0$ Substituting $$s=r-1$$ now gives: $4r(r-13)=0$ giving the radius of the circle for the head as 13 centimetres.