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the numbers $1,2...n$ are written at the vertices of an $n$-gon. One can add $1$ to the numbers at two adjacent vertices. The question is to determine for which $n$ can one apply this operation to obtain the same number at every vertex. It's not to hard to prove it can be done for all odd $n$, but can someone prove it can't be done when $n$ is even?

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Let $n$ be even and label the vertices of the $n$-gon clockwise, say. Then the alternating sum of the values of the vertices is $$\begin{align}1-2+3-4+\cdots+(n-1)-n &= (1-2) + (3-4) + \cdots + ((n-1)-n) \\ &= (-1) + (-1) + \cdots + (-1)\\ &= -\dfrac{n}{2} \end{align}$$ Furthermore, incrementing adjacent vertices does not change the value of the alternating sum, since no matter which two vertices we increment, we change the sum by $1-1=0$.

But if all the vertices had the same value then the alternating sum would be $0$. So this can't happen.

It can happen if you don't have to label the vertices (anti)clockwise, though. For instance: $$\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \to \begin{matrix} 2 & 3 \\ 3 & 4 \end{matrix} \to \begin{matrix} 3 & 3 \\ 4 & 4 \end{matrix} \to \begin{matrix} 4 & 4 \\ 4 & 4 \end{matrix}$$

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