Karel and Vojta found out that the kitchen clock at the cottage is fast by 1.5 minutes per hour and the clock in the bedroom is slow by half a minute per hour. At noon, they set both clocks to the same and correct time. The kitchen and bedroom clocks both have a standard twelve-hour dial. Determine when:

1. the kitchen clock will show the correct time again,
2. the bedroom clock will show the correct time again,
3. both clocks will show the same (though possibly incorrect) time again.

(M. Volfová)
1. The clock will show the correct time again when it runs ahead of the actual time by 12, 24, 36, ... hours. First, this will happen when it runs ahead by 12 hours, or 720 minutes. This will be achieved in $720: 1.5=480$ hours. The kitchen clock will show the correct time again for the first time after 480 hours (which is exactly 20 days).

2. The clock will show the correct time again for the first time when it is 12 hours, or 720 minutes, behind the actual time. This will be achieved in $720: 0.5=1440$ hours. The bedroom clock will show the correct time again for the first time after 1440 hours (which is exactly 60 days).

3. Every hour, the difference in time shown by the kitchen clock compared to the time shown by the bedroom clock increases by $1.5+0.5=2$ minutes. This difference must gradually reach 720 minutes, which will happen in $720: 2=360$ hours. The clocks will show the same time again for the first time after 360 hours (which is exactly 15 days).

Evaluation. Each part of the task is worth 2 points, with 1 point always for justification.