Problem of the Month
Hint for Problem 6: Regular Polygons and Lattice Points

1. Choose some lattice points $A$ and $B$ and compute $\tan (\mathrm{\angle }AOB)$. The values of $\tan (\mathrm{\angle }AOB)$ are limited to some subset of the real numbers. Can you figure out what this subset is, and can you prove that $\tan (60\mathrm{°})$ is not in that subset?

2. 1. Compute the interior angle of a regular pentagon. Then compute $\mathrm{\angle }CBF$. Is there a theorem from geometry you can use to prove that two lines are parallel?

   2. Suppose $X$ and $Y$ are points with coordinates $(x_1,x_2)$ and $(y_1,y_2)$. Let $Z$ be the point with coordinates $(x_1+y_1,x_2+y_2)$. Plot the four points $O,X,Y,Z$ on the Cartesian plane. Is there anything special about the quadrilateral with vertices $O,X,Y,Z$? Try it with some specific points $X$ and $Y$.

3. 1. Start by computing the angle between $L_1$ and $L_2$.

   2. Let $A$ be the center of the $n$-gon $B_1{B}_2\cdots B_n$ (that is, $A$ is the point where all the lines $L_i$ meet in the second image in the statement of the problem). Then $\mathrm{△}A{B}_1{B}_2$ is an isosceles triangle with one of its side lengths equal to $x$, and another one equal to $y$.

4. Use Question $1$ to rule out the existence of regular lattice triangles and hexagons. Use Question $2$ to rule out the existence of regular lattice pentagons. Use Question $3$ to rule out the existence of regular lattice $n$-gons where $n\ge 7$.

   As a general strategy, assume that there is a regular lattice $n$-gon, and try to construct a smaller regular lattice $n$-gon. If you can do this once, then you can do it again and again. Is it a problem to have smaller and smaller lattice polygons? Is this even possible?