Starting at midnight, Serge writes down the time after 1 minute, then 2 minutes after that, then 3 minutes after that, and so on. For example, the first four times that he writes are 12:01 a.m., 12:03 a.m., 12:06 a.m., and 12:10 a.m. He continues this process for 24 hours. What times does Serge write down that are exactly on the hour? (For example, 3:00 a.m. and 8:00 p.m. are exactly on the hour, while 11:57 p.m. is not.)
From the given information, Serge writes down the times that are $1+2+3+\cdots+(n-1)+n$ minutes after midnight for each successive value of $n$ from 1 up until the total is large enough to exceed 24 hours in minutes, which equals $24 \cdot 60=1440$ minutes.

Now $1+2+3+\cdots+(n-1)+n=\frac{1}{2} n(n+1)$ for each positive integer $n$.

To find times that are "on the hour", we find the values of $n$ for which $\frac{1}{2} n(n+1)$ is a multiple of 60 (that is, the time is a multiple of 60 minutes after midnight).

Since we want $\frac{1}{2} n(n+1) \leq 1440$, then $n(n+1) \leq 2880$.

Since $52 \cdot 53=2756$ and $53 \cdot 54=2862$ and $n(n+1)$ increases as $n$ increases, then we only need to check values of $n$ up to and including $n=52$.

For $\frac{1}{2} n(n+1)$ to be a multiple of 60 , we need $\frac{1}{2} n(n+1)=60 k$ for some integer $k$, or equivalently $n(n+1)=120 k$ for some integer $k$.

Note that $n$ and $n+1$ are either even and odd, or odd and even, respectively.

Note also that $120 k$ is divisible by 8 .

Thus, one of $n$ or $n+1$ is divisible by 8 .

Since the maximum possible value of $n$ is 52 , we can check the possibilities by hand:

| $n$ | $n(n+1)$ | Divisible by $120 ?$ | $\frac{1}{2} n(n+1)$ |
| :---: | :---: | :---: | :---: |
| 7 | 56 | No |  |
| 8 | 72 | No |  |
| 15 | 240 | Yes | 120 |
| 16 | 272 | No |  |
| 23 | 552 | No |  |
| 24 | 600 | Yes | 300 |
| 31 | 992 | No |  |
| 32 | 1056 | No |  |
| 39 | 1560 | Yes | 780 |
| 40 | 1640 | No |  |
| 47 | 2256 | No |  |
| 48 | 2352 | No |  |

(Note that we could have ruled out the cases $n=7,8,16,23,31,32,47,48$ by noting that in these cases neither $n$ nor $n+1$ is divisible by 5 and so $n(n+1)$ cannot be a multiple of 120 .) Therefore, the times that Serge writes down are those that are 120,300 and 780 minutes after midnight, which are 2, 5 and 13 hours, respectively, after midnight.

That is, Serge writes down 2:00 a.m., 5:00 a.m., and 1:00 p.m.

ANSWER: 2:00 a.m., 5:00 a.m., 1:00 p.m.