Equality in random configuration

If $n$ dice are rolled, the symmetry of the distribution for the total obtained means that the probability of rolling $X$ is the same as the probability of rolling $7n - X$ (and, moreover, $7n-X$ is the only other value that will be scored with the same probability as $X$ ). Thus the candidate value for $S$ is $7n-2018$ . We cannot score $2018$ by rolling $336$ dice or fewer (but we can if we roll between $337$ and $2018$ dice), and so the smallest possible value of $n$ is $337$ . To minimize $S$ , we must minimize $n$ , and so the answer is $7\times337 - 2018 = \boxed{341}$ .