3. Problem: You have an unlimited supply of square tiles with side length 1 and equilateral triangle tiles with side length 1 . For which $n$ can you use these tiles to create a convex $n$-sided polygon? The tiles must fit together without gaps and may not overlap.
Solution: All the angles in squares and equilateral triangles are multiples of $30^{\circ}$. So all the external angles of the $n$-sided polygon are multiples of $30^{\circ}$. Since the polygon is convex, this implies that all external angles are greater than or equal to $30^{\circ}$. However, the sum of the external angles is $360^{\circ}$, therefore
$$
n \times 30^{\circ} \leq 360^{\circ} .
$$

Hence $n \leq 12$. Also all polygons have at least 3 sides so $3 \leq n \leq 12$. Finally we demonstrate that it is possible for any $3 \leq n \leq 12$ using the following illustrations.