6. A factory produces sets of $n>2$ elephants of different sizes. According to the standard, the difference in mass between adjacent elephants within each set should be the same. The inspector checks the sets one by one using a balance scale without weights. For what smallest $n$ is this possible?
- Omвem: for $n=5$. Let's number the elephants in ascending order of their size, then it is enough to verify that $C_{1}+C_{4}=C_{2}+C_{3}, C_{1}+C_{5}=C_{2}+C_{4}$ and $C_{2}+$ $C_{5}=C_{3}+C_{4}$. These equalities are equivalent to $C_{4}-C_{3}=C_{2}-C_{1}$, $C_{2}-C_{1}=C_{5}-C_{4}$ and $C_{5}-C_{4}=C_{3}-C_{2}$, that is, all adjacent differences are equal. For $n=4$, it is impossible to check, because the weight sets $10,11,12,13$ and $10,11,13,14$ will give the same result for any weighings.

(A. Shapovalov)

## Third Round