Perpendicular hour and minute hand 
Starting at $12:00$ on a $12$-hour clock, how many times will the hour and minute hands be perpendicular to each other in a 12-hour period?

Each hour there are two instances when the clock has a $90°$ angle, so (naturally) I thought that the answer would be $12•2=24$, which is wrong. I have searched some answers before this, and one of them says that "as the hours progress, the minute hand lines up 90 degrees later in the hour", but I don't see how that interferes with the $12•2$ reasoning... no matter what the hour, the two 90° cases happen within the hour, don't they? 
 A: Sometimes a graph explains more than a thousand word.
On the horizontal axis I put the hours from $0$ to $12$ the purple line is the angle that the hour hand forms with the $00:00$ position.
On the vertical axis there is the angle that the minute hand forms with the $00:00$ position and is periodic since every hour the minute hand does the $360°$ round and then starts again from $0$. I have found the moments when the hands overlay and then shifting up and down by $90°$, $15$ minutes, the moments when the hands are perpendicular.
Keep in mind that the system is not isometric. To make a clearer drawing I used a $1:4$ scale. In other words four units on the $y-$axis is one hour like one unit on the $x-$axis.
If you want to know at what time the hands are perpendicular for the first time you solve the system formed by the equations of the two lines: $y=x$ the line of the minute hand and $y=\dfrac{x}{12}+\dfrac{1}{4}$ the hour hand shifted of $90°$ (=fifteen minutes). We find $\left(\frac{3}{11},\;\frac{3}{11}\right)$ which means $3/11+3/11\times 60\approx 00:16:38.2$ that is noon sixteen minutes and $38.2$ seconds.
In a similar way can be found all the other moments, changing a little the equation
hope it helps

A: In each time interval $[n, n+1]$ hours, the minute hand is perpendicular to the hour hand twice, and there are 12 of these intervals. However, since the hands are perpendicular at 3:00 and 9:00, which each belong to two of these intervals, we overcounted by 2. Thus, the hour and minute hands are perpendicular 22 times.
