A clock strikes once at 1 o'clock, twice at 2 o'clock, ..., twelve times at 12 o'clock. How many times will the bell strike in the course of $2$ days?

First, find the number of strikes in a single $12$ -hour cycle. This is the sum of the first $12$ natural numbers:
$$\text{Sum}=1+2+3+\cdots +12=\frac{12\times (12+1)}{2}=6\times 13=78\text{ strikes}$$

A single day has $24$ hours, which comprises two $12$ -hour cycles.
Therefore, $2$ days contain a total of:

$$\text{Total cycles}=2\text{ days}\times 2\text{ cycles/day}=4\text{ cycles}$$

Total number of strikes over $2$ days:

$$\text{Total strikes}=78\times 4=312$$