Solution:

The answer is (D). The goals scored by the physicists in the first $n$ matches are equal to the sum of the first $n$ odd numbers, that is, they are $n^{2}$. Since the mathematicians won, they scored more than $330/2 = 165$ goals, so the physicists scored fewer than $165$ goals. The largest perfect square less than $165$ is $144=12^{2}$, which represents the maximum number of goals that the physicists could have scored. Hence the mathematicians scored at least $330-144=186$ goals and the minimum margin is therefore $186-144=42$, which could have been achieved if, for example, the mathematicians scored $15$ goals in $6$ of the $12$ matches and $16$ goals in the remaining $6$.