2. (1988 US Mathcounts Math Competition) In a game, answering an easy question scores 3 points, and answering a hard question scores 7 points. Among the set of integers that cannot be the total score of a contestant, find the maximum value.

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2. Let the set of natural numbers that cannot be used as total scores be $S$. Among the first 12 natural numbers, only $1,2,4,5,8,11$ belong to $S$.

Since $7=2 \times 3+1,3 \times 5=7 \times 2+1$, when the sum of several 3s and 7s $n \geqslant 12$, one can replace a 7 with two 3s, or replace five 3s with two 7s, to increase the sum by 1. Therefore, natural numbers greater than 12 do not belong to $S$. So
$$
S=\{1,2,4,5,8,11\} \text {. }
$$

The maximum value sought is 11.