It is currently $3\!:\!00\!:\!00 \text{ p.m.}$ What time will it be in $6666$ seconds?  (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)
There are $60$ seconds in a minute. When $6666$ is divided by $60$, you get $111$ with a remainder of $6$ seconds. Therefore, $6666$ seconds is $111$ minutes and $6$ seconds. There are $60$ minutes in an hour. When you divide $111$ by $60$, you get $1$ with a remainder of $51$. Thus, $6666$ seconds is equivalent to $1$ hour $51$ minutes and $6$ seconds. Therefore, the time in $6666$ seconds is $\boxed{4\!:\!51\!:\!06 \text{ p.m.}}$
4\!:\!51\!:\!06p.m.
Level 3
Number Theory
Problem: It is currently $3\!:\!00\!:\!00 \text{ p.m.}$ What time will it be in $6666$ seconds?  (Enter the time in the format "HH:MM:SS", without including "am" or "pm".)
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the time after $6666$ seconds from $3\!:\!00\!:\!00$, first convert $6666$ seconds into hours, minutes, and seconds. 

1. Calculate the total number of seconds in an hour: 
   1 hour = $3600$ seconds.

2. Determine how many complete hours are in $6666$ seconds:
   $6666 \div 3600 \approx 1.85166667$ hours. 
   This means there is $1$ complete hour.

3. Calculate the remaining seconds after accounting for $1$ hour:
   $6666 - 3600 = 3066$ seconds.

4. Next, convert $3066$ seconds into minutes:
   1 minute = $60$ seconds.
   $3066 \div 60 \approx 51.1$ minutes.
   This means there are $51$ complete minutes.

5. Calculate the remaining seconds after accounting for $51$ minutes:
   $3066 - (51 \times 60) = 3066 - 3060 = 6$ seconds.

6. Now, add the calculated time to the original time:
   Start with $3\!:\!00\!:\!00$.
   Adding $1$ hour gives $4\!:\!00\!:\!00$.
   Adding $51$ minutes gives $4\!:\!51\!:\!00$.
   Finally, adding $6$ seconds gives $4\!:\!51\!:\!06$.

7. The final answer is formatted as required, resulting in $4\!:\!51\!:\!06p.m.$.<|END_OF_LATENT|> The final answer is $4\!:\!51\!:\!06p.m.$.