Several weights are given, each of which is not heavier than $1 \mathrm{lb}$. It is known that they cannot be divided into two groups such that the weight of each group is greater than $1 \mathrm{lb}$. Find the maximum possible total weight of these weights.
Suppose you start putting weights into two pans of a scale and stop when you cannot add another weight to either pan without it exceeding $1 \mathrm{lb}$. At this point, you have put at most $2 \mathrm{lb}$ of weight on the scale, and there can only be one weight left, otherwise you could add a weight to each pan and both would exceed $1 \mathrm{lb}$. This remaining weight weighs at most $1 \mathrm{lb}$, so the weights add up to at most $3 \mathrm{lb}$ total. It is possible to have exactly $3 \mathrm{lb}$ with three $1 \mathrm{lb}$ weights, so this is the largest possible.