11. When $n$ fair dice are thrown, the probability of obtaining a sum of 2005 and the probability of obtaining a sum of $S$ are both equal to a positive number $p$. Find the smallest possible value of $S$.
(1 mark)
投擲 $n$ 顆公平的骰子時，擲得點數總和為 2005 及點數總和為 $S$ 的概率皆等於某個正數 $p$ 。求 $S$ 的最小可能值。
11. 340
11. When $n$ dice are thrown, the smallest possible sum obtained is $n$ and the greatest possible sum obtained is $6 n$. The probabilities of obtaining these sums are symmetric about the middle, namely, $\frac{7 n}{2}$. In other words, the probabilities of obtaining a sum of $S$ and obtaining a sum of $7 n-S$ are the same. (To see this, we may consider the symmetry $1 \leftrightarrow 6,2 \leftrightarrow 5,3 \leftrightarrow 4$.) Furthermore, the probability for obtaining the different possible sums increases for sums from $n$ to $\frac{7 n}{2}$ and decreases for sums from $\frac{7 n}{2}$ to $6 n$. Thus the same probability occurs for at most two possible sums. From the above discussions, we know that $S$ is either equal to 2005 or $7 n-2005$. Therefore, to minimise $S$ we should minimise $n$.

Since $2005=6 \times 334+1, n$ is at least 335 . Indeed, when $n$ is $335, S$ may be equal to $7 n-2005=7(335)-2005=340$. Since 340 is smaller than 2005 , we conclude that the smallest possible value of $S$ is 340 .