## Task 3 - 340823

a) How many times in total do the hour and minute hands of a clock stand perpendicular to each other over the course of 24 hours (from 0:00 to 24:00)?

b) In particular, calculate all such times between 4:00 and 5:00! Give these times as they would be displayed on a digital clock, assuming that the clock is accurate, i.e., it shows the correct hour, minute, and second at the start of each second!
(a) Since both hands move at constant speeds, the following applies:

If the hands overlap exactly $n$ times during the course of 24 hours (with the beginning of this time span being included but not the end), then there are also exactly $n$ times in this period when the minute hand is $90^{\circ}$ ahead of the hour hand, and exactly $n$ times when the minute hand is $90^{\circ}$ behind the hour hand.

This number $n$ can be found as follows:

In 12 hours, the minute hand makes exactly 12 complete revolutions, while the hour hand makes exactly one revolution. Therefore, there must be exactly 11 times in this period when the minute hand overtakes the hour hand. In 24 hours, there are twice as many such times; hence, $n=22$.

Thus, the hands are perpendicular to each other a total of 44 times in the course of 24 hours.

(b) At 4 o'clock, the hour hand is 20 divisions ahead of the minute hand. If this distance has decreased by 5 divisions for the first time after $x$ minutes, this is because the minute hand has moved forward by $x$ divisions, while the hour hand has moved forward by only $\frac{x}{12}$ divisions. Therefore, $x=5+\frac{x}{12}$.

This results in $12x=60+x, x=5\frac{5}{11}$; i.e., the first desired time is $5\frac{5}{11}$ minutes after 4 o'clock. In digital format, this time is displayed as 04:05:27.

If, however, $y$ minutes after 4 o'clock, the minute hand has overtaken the hour hand, thus having covered the original distance of 20 divisions and is now 15 divisions ahead of the hour hand for the first time, then $y=20+15+\frac{y}{12}$ with $y=38\frac{2}{11}$. For the second desired time, the digital display is 04:38:10.