TITLE: On a Putnam's 2009 problem
QUESTION [14 upvotes]: Find all even natural numbers $n$ such that the following is true: 
There is a non-constant function $f : \Bbb{R}^2 \longrightarrow \Bbb{Z}_2$ such that for any regular $n$-gon $A_1...A_n$, $f(A_1) + \cdots + f(A_n) =0$. 
NOTE: This is the unsolved part of THIS  problem. I chose to state it in a separate question first to get more attention and second since some people may want to put a bounty on this part.

REPLY [2 votes]: Below are pictures that prove the $n=8$ and make progress on the $n=10$ case. I think this is very close to a complete argument for all $n$. For the left hand picture there are 2 black $8$-gons and $4$ green $8$-gons. Most points get hit twice. The origin gets hit 4 times. What's left over are the $4$ vertices of a square. Thus we have shown that the sum over vertices of a square is zero, so we have reduced to a known case. For the second picture, there are 2 black $10$-gons and $5$ green $10$-gons. Summing over everything gives the sum of the vertices of a $5$-gon plus its center is zero. One can probably go further with this, but my wife will get angry with me if I spend any more time on it today.