Infinitely many math beasts stand in a line, all six feet apart, wearing masks, and with clean hands. Grogg starts at the front of the line, holding $n$ pieces of candy, $ n \ge 1,$ and everyone else has none. He passes his candy to the beasts behind him, one piece each to the next $n$ beasts in line. Then, Grogg leaves the line. The other beasts repeat this process: the beast in front, who has $k$ pieces of candy, passes one piece each to the next $k$ beasts in line, and then leaves the line. For some values of $n,$ another beast, besides Grogg, temporarily holds all the candy. For which values of $n$ does this occur?
1. **Initial Setup and Definitions:**
   - Grogg starts with $n$ pieces of candy.
   - He distributes one piece of candy to each of the next $n$ beasts in line.
   - Grogg then leaves the line.
   - Each subsequent beast in the front of the line, who has $k$ pieces of candy, distributes one piece to each of the next $k$ beasts in line and then leaves the line.

2. **Claim:**
   - For some values of $n$, another beast, besides Grogg, temporarily holds all the candy.

3. **Base Case:**
   - Consider $n = 1$:
     - Grogg gives his one piece of candy to the next beast.
     - The next beast now has all the candy.
     - Thus, for $n = 1$, another beast temporarily holds all the candy.

4. **Inductive Step:**
   - Consider $n = 2$:
     - Grogg gives one piece of candy to each of the next two beasts.
     - The distribution is now: $1, 1$.
     - The first beast in line gives its one piece of candy to the next beast.
     - The distribution is now: $2$.
     - Thus, for $n = 2$, another beast temporarily holds all the candy.

5. **General Case for $n > 2$:**
   - Assume $n > 2$ and consider the distribution after Grogg distributes his $n$ pieces of candy.
   - The distribution is: $1, 1, 1, \ldots, 1$ (with $n$ ones).
   - The first beast in line gives its one piece of candy to the next beast.
   - The distribution is now: $1, 1, 1, \ldots, 1$ (with $n-1$ ones).
   - This process continues, and each beast in turn gives its one piece of candy to the next beast.
   - At no point does any single beast hold all $n$ pieces of candy simultaneously.

6. **Conclusion:**
   - For $n > 2$, no beast other than Grogg temporarily holds all the candy.
   - For $n = 1$ and $n = 2$, another beast temporarily holds all the candy.

The final answer is $\boxed{n = 1, 2}$