54. At a Right Angle. Once during breakfast, Professor Rackbrain asked his young friends a simple question:

- When between five and six o'clock will the hour and minute hands be exactly at a right angle?
54. For the angle between the hands to be a right angle, the minute hand must be exactly 15 minutes ahead of or behind the hour hand. Each of these positions will occur 11 times in 12 hours, that is, every 1 hour \(5 \frac{5}{11}\) minutes. If eight such intervals pass after 9 o'clock, the clock will show 5 hours \(43 \frac{7}{11}\) minutes. On the other hand, if two such intervals pass after 3 o'clock, we get 5 hours \(10 \frac{10}{11}\) minutes. These are the two moments in time,
\footnotetext{
* Here M. Gardner is not entirely correct, as Dudeney considers IX as a combination of two digits: I and X.
}
that were required to find in the problem, with the second moment, of course, occurring earlier than the first.