Artificial intelligent assistant

regular k-gons that can tessellate a plane > What is the sum of all possible values of $k$ for which regular $k-$gons can tessellate a plane? This is one of the NAT problems. While I am familiar with tessellations, I don't quite get what they mean by tessellating a plane. Shouldn't all polygons be able to fill up a plane?

An $n$-gon has angle sum $(n-2)180^\circ$. If this $n$-gon is regular each of its angles therefore is ${n-2\over n}180^\circ$. Since we want that an integer number $k$ of such regular $n$-gons meet at each vertex of the tessellation we have to insist that ${n-2\over n}180^\circ={1\over k}360^\circ$, or $$k={n\over n-2}\in{\mathbb N}\ .$$ Now try $n=3$, $4$, $5$, $\ldots$, and see in which cases the resulting $k$ is integer. You will obtain a finite list of admissible $n$s and then have to check for which of these $n$s a tessellation is indeed possible.

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