Solution to Riddle of the Week #33: The Hourglass Problem

This problem is similar in spirit to the notoriously tricky Burning Rope Problem. The key is to figure out how to use your two time-measuring devices in such a way that they compliment each other. You can see the original problem here.

Solution

Here is the first way to measure out 15 minutes:

Flip both the 11-minute and the 7-minute hourglasses over to start them both.
When the 7-minute hourglass runs out, that is the moment when you start to measure the target 15 minutes.
Four minutes after the 7-minute hourglass runs out, the 11-minute hourglass will run out.
Flip the 11-minute hourglass over immediately to measure another 11 minutes, landing you at 15 minutes exactly.

But there is a better way! The solution above requires 22 minutes, but it is possible to measure the 15 minutes in just 15 minutes without any wasted time. The second solution:

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Flip both the 11-minute and the 7-minute hourglasses over to start them both.
When the 7-minute hourglass runs out, flip it over immediately to restart it.
When the 11-minute hourglass runs out, flip the 7-minute hourglass over again, even though it is only partly finished.
When the 7-minute hourglass runs out, exactly 15 minutes have passed.

Let's take a moment to explain the second solution. You flip them both, and then when the 7-minute glass runs out, 7 minutes have passed. You flip the 7-minute glass over again to restart it. Then, four minutes later, the 11-minute hourglass runs out. That means the total time elapsed is 11 minutes. It also means that the 7-minute hourglass has 3 minutes remaining—and, it means that if you flip it over at that moment, there will be four minutes left until it runs out. Because 11 minutes have passed and the 7-minute hourglass will now measure out just four more minutes, you get exactly 15 minutes when it runs out.

Come back next week for another riddle!

*See all of our riddles here.