How Many Elves are There?

After $1$ hour the $1$ elf makes $2$ new elves for a total of $1 + 2 = 3$ elves, after $2$ hours these $3$ elves make $6$ new elves for a total of $3 + 6 = 9$ elves, and so on.

Generally, for $n$ hours and $e_n$ elves, after $n$ hours $e_{n - 1}$ elves make $2e_{n - 1}$ new elves for a total of $e_{n - 1} + 2e_{n - 1} = 3e_{n - 1}$ elves, which means $e_n = 3e_{n - 1}$ , which can inductively be shown to be $e_n = 3^n$ .

Therefore, after $24$ hours, there are $e_{24} = \boxed{3^{24}}$ elves.

Let the total number of elves at $n$ hours later be $E_n$ . Since each of the $(n-1)$ th elves makes two elves, $E_n = E_{n-1} + 2E_{n-1} = 3E_{n-1}$ . And since the starting number of elves $E_0 = 1$ , $E_1 = 3E_0 = 3$ , $E_2 = 3E_1 = 3^2$ , $E_3 = 3E_2 = 3^3 \cdots$ , $\implies E_n = 3^n$ , $\implies E_{24} = \boxed{3^{24}}$ .