A square tile, a regular hexagon tile, and a regular $n$-sided polygon tile share a common vertex. There are no gaps or overlaps between the tiles. What is $n?$


The exterior angles of a $k$-sided polygon form an arithmetic sequence. The smallest and largest interior angles of the polygon are $136^{\circ}$ and $176^{\circ}$. What is $k?$

 Feb 5, 2021

1.  The sum of the angles surrounding the common vertex is $360^\circ$. The right angle from the square corner is $90^\circ$ of that, and the interior angle of the hexagon is another $120^\circ$. Thus, the remaining angle is $360^\circ-90^\circ-120^\circ=150^\circ$. Since $150^{\circ}$ is an interior angle of the $n$-gon, each exterior angle measures $30^{\circ}$. Thus, the number of vertices is $\dfrac{360^{\circ}}{30^{\circ}}=\boxed{12}$.


2.  The largest and smallest exterior angles measure $44^{\circ}$ and $4^{\circ}$. Since the exterior angles are in an arithmetic progression, the average of all of them is equal to the average of the largest and smallest. Since we have $k$ angles in total, and their sum is $360^{\circ},$ this means $$\dfrac{360^{\circ}}{k}=\dfrac{44^{\circ}+4^{\circ}}{2}=24^{\circ},$$so $k=\dfrac{360^{\circ}}{24^{\circ}}=\boxed{15}$.

 Feb 5, 2021

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