A tile, in the shape of a regular polygon, was removed from the place it occupied on a panel. It was then observed that if this tile were to undergo a rotation of $40^{\circ}$ or $60^{\circ}$ around its center, it could be perfectly fitted into the space left vacant on the panel. What is the smallest number of sides that this polygon can have?

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Solution

For such rotations to be possible, with the fitting of the polygon, it is necessary and sufficient that the central angle be a divisor of $40^{\circ}$ and $60^{\circ}$. If $n$ is the number of sides of the tile, the central angle is given by $\frac{360^{\circ}}{n}$. Thus, the ratios

$$
\frac{40^{\circ}}{\frac{360^{\circ}}{n}}=\frac{n}{9} \text{ and } \frac{60^{\circ}}{\frac{360^{\circ}}{n}}=\frac{n}{6}
$$

must be integers. The smallest positive integer that is a multiple of both 6 and 9 is 18. Clearly, a regular polygon with 18 sides, having a central angle of $20^{\circ}$, satisfies the condition stated in the problem.