62nd Czech and Slovak Mathematical Olympiad

Let $b$ be the sum of numbers on side faces of the pyramid, let $a$ be the number on the base. After a step involving any base vertex, $b$ increases or decreases by $2$ and $a$ increases or decreases by $1$, that means the value $V=b-2a$ stays the same. If we choose for a step the apex, only $b$ increases or decreases by $n$, thus $V$ increases or decreases by $n$ as well. Therefore $V$ is in the process always divisible by $n$. But in the position with number $1$ written on each side, the corresponding $V$ is $n-2$, which is not divisible by $n$ (as $n>2$).