Nine Minutes from a 4 and a 7
The problem
You have two hourglasses: one runs for exactly minutes, the other for exactly minutes. You may flip either glass at any instant, including mid-run. You have no clock. How do you time an interval of exactly minutes?
- A glass that has just been flipped runs its full duration; a glass flipped mid-run restarts from the amount of sand already in the bottom.
- You have no clock, no ruler and no way to mark the sand.
Take a minute
Reveal hint 1
Start both glasses together. Once the -glass is flipped every time it empties, it fires on a strict -minute beat that is completely independent of the -glass.
Reveal hint 2
Nine minutes is . Can you recognise the moments and ?
Solution
AnswerStart both glasses at once. Begin timing the moment the -minute glass first empties, and flip the -minute glass each time it empties. Stop when the -minute glass empties for the third time after that.
Derivation
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Start both glasses at .
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Let the -glass keep its beat.
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Start the egg at the first emptying of the -glass.
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Count three emptyings of the -glass.
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Stop the egg at the third one.
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Why nothing is wasted.