You have two hourglasses: one for 7 minutes and one for 11 minutes. How can you measure exactly 15 minutes?
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The hourglasses (7 and 11 minutes)
Hints
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- Sequence: At t=7, the 7 is emptied. Turn it over immediately.
- Sequence: At t=11, the 11 is emptied. At that moment the 7 rotates again.
- Why step 3 works: upon reaching t=11, the 7 clock had been running for 4 minutes since its last spin, so when you turn it over there is exactly sand left for 4 more minutes.
Solution
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Answer: You can measure exactly 15 minutes.
Sequence:
- At $t=0$, start both clocks (7 and 11).
- In $t=7$, the 7 is emptied. Turn it over immediately.
- In $t=11$, the 11 is emptied. At that moment the 7 rotates again.
- The 7 clock is emptied again at $t=15$.
Why step 3 works: When you reach $t=11$, the 7's clock had been running for 4 minutes since its last turn, so when you turn it over there is exactly sand left for 4 more minutes.
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