## 6. Coffee is served

Coffee is always served to me between 1 and 2 o'clock, at a moment when the bisector of the angle formed by the two hands of my clock points exactly at the "12" (hour) mark. At what exact time is my coffee served?
6. Let's denote this unknown moment of time as 1 hour $x$ minutes (since it is between 1 hour and 2 hours!). The angle formed by the hour hand with the bisector of the angle between the two hands is (in degrees)

$$
\frac{360}{12}+\frac{x}{60} \cdot \frac{360}{12}=30+\frac{x}{2}
$$

(here we use the fact that the bisector of the angle points to the 12 o'clock mark).

The angle formed by the minute hand with this same bisector is

$$
360-x \cdot(360 / 60)=360-6 x
$$

${ }^{1}$ In Venn diagrams (cf. footnote on p. 119), each subset $Z$ is symbolically represented by a circle, making it easy to trace all possible unions and intersections of these sets, - Note by the translator.

Thus, we have

$$
30+x / 2=360-6 x
$$

from which

$$
x=(2 / 13) \cdot 300 \approx 50 \text { min } 46 \text{s}.
$$

Therefore, my coffee is served at approximately 1 hour 50 minutes $46 \text{s}$.