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Failure
Q: How can you use a 4-minute hourglass and a 7-minute hourglass to measure 9 minutes?
A: Here's a way to measure exactly 9 minutes using a 4-minute hourglass and a 7-minute hourglass:
Start both hourglasses at the same time. When the 4-minute hourglass runs out, flip it over (4 minutes elapsed). When the 7-minute hourglass runs out, flip the 4-minute hourglass again (7 minutes elapsed). The 4-minute hourglass had 1 minute left when you flipped it, so it will run for an additional minute. When the 4-minute hourglass runs out this time, 2 additional minutes have passed (for a total of 9 minutes elapsed). So, you've just measured exactly 9 minutes using a 4-minute hourglass and a 7-minute hourglass.
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Success
Q: How can you use a 4-minute hourglass and a 7-minute hourglass to measure 9 minutes? Please reason step-by-step. You are allowed to run preparatory steps beforehand if necessary.
A: The problem can be solved by using the following steps:
Preparatory Steps:
Start both hourglasses at the same time. When the 4-minute hourglass runs out, turn it over to start it again. (This marks 4 minutes.) When the 7-minute hourglass runs out, don't flip it yet, but note that the 4-minute hourglass has now been running for 3 minutes on its second run. (This marks 7 minutes.) When the 4-minute hourglass runs out again, flip the 7-minute hourglass. (This marks 8 minutes.)
Measurement Steps:
When the sand in the 7-minute hourglass runs out again, it will have been 1 minute since its flip, marking a total of 9 minutes. So, the process will be: 0-4 minutes: first run of the 4-minute hourglass 4-7 minutes: second run of the 4-minute hourglass until the 7-minute hourglass runs out 7-8 minutes: rest of the second run of the 4-minute hourglass 8-9 minutes: one minute of the 7-minute hourglass after flipping it when the 4-minute hourglass runs out the second time.
Thus, the total time measured is 9 minutes.