Pigeonhole and Extremal Arguments · Challenge problem

Choose $n$ different numbers from $\{1, 2, \ldots, 50\}$. Counting shows two different subsets must have the same sum once there are more non-empty subsets ($2^n - 1$) than possible sums (from $1$ up to the largest possible total). What is the smallest $n$ for which this count works for EVERY choice?