# Sunday Times Teaser 2572 – Abominable

### by Angela Newing

#### Published: 8 January 2012 (link)

In the art class Pat used five circles for his new design, Snowman. One circle made the head and another touching circle (with radius three times as big) made the body. Two equal circles (of radius one centimetre less than that of the head) made the arms (one on each side) and they touched both the body and the head. The fifth circle enclosed the other four, touching each of them. The radius of the head’s circle was a whole number of centimetres.

How many centimetres?

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1.  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.