Shapovadov A.V.

In a set of several weights, all of which have different masses. It is known that if any pair of weights is placed on the left pan, the scales can be balanced by placing one or several weights from the remaining ones on the right pan. Find the smallest possible number of weights in the set.
To balance a pair of the heaviest weights, at least three weights are needed, which means there are at least five weights in total. Suppose there are exactly five weights, and their weights are

$P_{1}2$, then a pair

$(m+1, n-1)$ weighs the same. If $m>3$ and $n<8$, then a pair ( $m-1, n+1$ ) weighs the same. The cases considered do not cover four pairs: $(3,4),(3,5),(6,8)$ and ( 7,8 ). They are balanced respectively by the sets (7), $(8),(3,4,7),(4,5,6)$.

## Answer

6 weights.