the observer and the times


2 hours and 12 minutes had passed.

The hour hand is exactly on a minute mark at five times during each hour – on the hour and at 12, 24, 36 and 48 minutes past.

Let x be the number of hours, and y the number of minutes past the hour. When the hour hand is on a minute mark, the position of the hour hand is 5x + y/12 and the position of the minute hand is y.

On the first occasion, y = 5x + y/12 + 6. This is equivalent to the equation 60x = 11y -72. Since y can only take one of the values 0, 12, 24, 36 or 48, it is easily determined that the only suitable values for the equation are: x = 1, y = 12. So the time is 1:12.

Similarly, on the second occasion 60x = 11y – 84. Here the only suitable values are: x = 3, y = 24. So the time is 3:24.