The Careless Watchmaker

Strategies and Hints

1. During the first hour the minute hand moves from 12 to 1. The hour hand makes a complete circle from 2 to 2. At 3:00 the watch shows 2:05. During the first hour the watch is continuously wrong.

2. During the second hour the minute hand moves from 1 to 2. The hour hand again makes a complete circle from 2 to 2. At 4:00 the watch shows 2:10. Yes, during the second hour the watch will show the correct time. At around 5 minutes past 3 the hour hand will be on the 3, and the minute hand will have moved one-twelfth of the distance from the 1 to the 2.

3. d = 12D

4. At the correct time of 3:00, the minute hand of the incorrect clock is +5 units from where it should be, and the hour hand is -5 units from where it should be. The hour hand moves 12 units for each 1 unit the minute hand moves. So the hour hand moves 12(d - 5) while the minute hand moves D + 5. The second equation is D + 5 = 12(d- 5).
Solution

Solving the system of equations d = 12D and D + 5 = 12(d - 5), gives the solution. The watch shows the correct time at minutes past three.