168. Inscribed Dodecagon. For a convex dodecagon inscribed in a circle, the lengths of some six sides are $\sqrt{2}$, and the length of each of the remaining six sides is $\sqrt{24}$. What is the radius of the circle?

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168. Let $A B$ and $B C$ be two adjacent but unequal sides of our dodecagon. Then $\smile A B+\smile B C=\smile A C=\frac{360^{\circ}}{6}=$ $60^{\circ}$; hence, $A C=r$. Applying the Law of Cosines to triangle $A B C$, where angle $A B C=150^{\circ}$, we get

$$
(A C)^{2}=(\sqrt{2})^{2}+(\sqrt{24})^{2}-2 \sqrt{2} \cdot \sqrt{24}\left(-\frac{\sqrt{3}}{2}\right)=38
$$

Thus, $r=\sqrt{38}$.

[Norman Anning, M. M., 28, 113 (November 1954).]