Question 142
At a market, a buyer and a seller each have four exotic coins. You are allowed to label each of the eight coins with any positive integer value in cents. The labeling is called $n$-efficient if for any integer $k$, $1 \leq k \leq n$, it is possible for the buyer and the seller to give each other some of their coins in such a way that, as a net result, the buyer has paid the seller $k$ cents. Find the greatest positive integer $n$ such that an $n$-efficient labeling of the coins exists.