Timing 9 Minutes with Sand-Watches
Introduction
Dr. Johnson possesses two sand-watches: a 7-minute and a 4-minute timer. The challenge is to accurately measure a total of 9 minutes using these devices.
Step-by-Step Solution
To achieve this, Dr. Johnson should follow these steps:
1. **Start Both Sand-Watches**: Begin by flipping both the 7-minute and 4-minute sand-watches simultaneously.
2. **Monitor the 4-Minute Sand-Watch**: When the 4-minute sand-watch runs out, flip it over immediately.
3. **Track the 7-Minute Sand-Watch**: Once the 7-minute sand-watch completes its cycle, turn it upside down right away.
4. **Final Minute Calculation**: At this point, the 4-minute sand-watch will have 1 minute of sand remaining. When it runs out, there will be 1 minute of sand still in the 7-minute sand-watch.
5. **Complete the Timing**: Flip the 7-minute sand-watch again and let the last minute run out.
Conclusion
The total time measured is 4 minutes (from the first 4-minute watch), plus 3 minutes (from the 7-minute watch), plus 1 minute (remaining in the 4-minute watch), and finally 1 minute (from the second flip of the 7-minute watch). This adds up to a total of 9 minutes, successfully timed with the sand-watches.