The honey cake maker, Master Bonifác, loved to bring joy to as many children as possible, which is why he was constantly active when awake. If he slept little, he worked drowsily. Excessive sleep also made him feel sluggish, and he had less time to work. He realized that the number of cakes baked per hour was proportional to the product of the duration of his sleep and wakefulness. How many hours should Master Bonifác sleep each day to be able to bake the most cakes?
If a master bakes $x$ loaves of bread in 1 hour and works $t$ hours a day, and rests for $(24-t)$ hours, then according to the condition,

$$
\frac{x}{t(24-t)}=k \quad(k \text { is the proportionality factor }) .
$$

From this, $x=k t(24-t)$, which is the number of loaves baked in 1 hour.

If he works $t$ hours, then he bakes $f(t)=k t^{2}(24-t)$ amount. We need to find the maximum of this function. Let's write down the derivative function:

$$
f^{\prime}(t)=k\left(48 t-3 t^{2}\right)=k t(48-3 t)
$$

The function can have extreme values at points where the derivative function is 0, except at $t=0$ and $t=24$.

This latter is satisfied if $t=0$ (which is not of interest to us), or if $t=16$.

Since the function $f$ is monotonically increasing in the interval $(0,16)$, 0 at 16, and decreasing in the interval $(16,+\infty)$, it indeed has a maximum at $t=16$.

Therefore, Master Bonifác should work 16 hours a day to bake the most loaves of bread.

Remark. The inequality between the arithmetic and geometric means also leads to the goal. According to this,

$$
f(t)=4 k\left(\frac{t}{2} \cdot \frac{t}{2} \cdot(24-t)\right) \leq 4 k\left(\frac{\frac{t}{2}+\frac{t}{2}+(24-t)}{3}\right)^{3}=4 k \cdot 8^{3}
$$

and equality holds precisely when the factors $\frac{t}{2}, 24-t$ are equal, i.e., $t=16$.