47. This happens at 9:06 \(\frac{3}{4}\) when the hour hand has traveled a distance of \(45 \frac{9}{16}\) (which is \(6 \frac{3}{4}\) squared) minute divisions (after XII). If we allowed fractions less than one minute, there would be another solution, namely: 12:05 ( \(\frac{1}{12}\) min).
47. When does this happen? When the clock hands are arranged in such a way that if the distance is taken as the number of minute divisions after XII, then the path traveled by one of the hands is equal to the square of the path traveled by the other?