Question 139
An equilateral triangle is divided into 9 small triangles as in the figure, and on each small triangle the number 0 is initially written. Marco, to pass the time, plays the following game: at each move he chooses 2 small triangles sharing a side and adds 1 to, or subtracts 1 from, both numbers written on these small triangles (it is understood that the operation performed on the two small triangles is the same). After some time he notices that the numbers written on the 9 small triangles are, in some order, $n, n+1, \ldots, n+8$, where $n$ is a non-negative integer. Prove that $n$ can only be 0 or 2.
Note. The cases $n=0$ and $n=2$ can actually occur, but you are not asked to prove this.
Note. The cases $n=0$ and $n=2$ can actually occur, but you are not asked to prove this.