## 17. Mathematical Puzzle

This week's mathematical puzzle in the "Modern Values" magazine was particularly difficult. When I started thinking about it (it was between 9 and 10 o'clock), I noticed that the two hands of my clock were in a straight line, one continuing the other. And I found the solution only at the moment when the hands merged with each other.

How long did my reflections last?
17. Let 9 hours $x$ minutes be the time when I started solving the problem. The angle formed by the hour hand with the common direction of the clock hands at noon is (in degrees)

$$
2 \cdot 360 / 12 + [(60 - x) / 60] \cdot 30
$$

The angle formed by the minute hand with the same direction is

$$
(360 / 60) \cdot x
$$

Since the hands were extensions of each other, these two angles must be supplementary to $180^{\circ}$:

$$
60 + (60 - x) / 2 + 6x = 180 \text{ or } 30 + (11 / 2) \cdot x = 120
$$

from which

$$
x = (2 / 11) \cdot 90 \approx 16 \text{ min } 22 \text{s}
$$

Let 10 hours minus $y$ minutes be the time when I finished my thoughts. The angle formed by the hour hand with the common direction of the clock hands at noon is (in degrees)

$$
2 \cdot (360 / 12) + (y / 60) \cdot 30
$$

The angle formed by the minute hand with the same direction is

$$
(360 / 60) \cdot y
$$

Since the two hands merged, these angles must be equal to each other

$$
6y = 60 + y / 2
$$

from which

$$
y = 120 / 11 \approx 10 \text{ min } 54 \text{s}.
$$

Thus, I started working at approximately 9:16:22, and finished at approximately 9:49:06. My thoughts on the puzzle took me about 32 minutes and 45 seconds.