Measuring Time with Hourglasses 3
Previous puzzle in this series
As your cooking skills improve, you decide to try preparing baked salmon fillet. It needs to go into the oven for exactly 20 minutes. As before, you have two hourglasses: one measures 7 minutes, the other measures 9 minutes. Using only these two hourglasses, how can you measure exactly 20 minutes?
Hint
Try solving Measuring Time with Hourglasses 2 first.
Hint
You need a preparation phase. Run the hourglasses for a while to reach a moment where you can start timing, and then a later moment exactly 20 minutes after.
Solution
Unlike the earlier hourglass puzzles, you can’t begin measuring right away. No matter what combination of moves you try, you can’t measure exactly 20 minutes. You need a preparation phase: start flipping, and then pick the right moment to begin and end your measurement.
Here’s how it plays out. Start both hourglasses at the same time.
7-min hourglass: 7 minutes left
9-min hourglass: 9 minutes left
Elapsed: 0 min
At 7 minutes, the 7-minute hourglass runs out. Flip it as you start the oven.
7-min hourglass: 7 minutes left
9-min hourglass: 2 minutes left
Elapsed: 7 min — start timing
At 9 minutes, the 9-minute hourglass runs out. Flip it.
7-min hourglass: 5 minutes left
9-min hourglass: 9 minutes left
Elapsed: 9 min (2 min measured)
At 14 minutes, the 7-minute hourglass runs out. Flip it.
7-min hourglass: 7 minutes left
9-min hourglass: 4 minutes left
Elapsed: 14 min (7 min measured)
At 18 minutes, the 9-minute hourglass runs out. Flip it.
7-min hourglass: 3 minutes left
9-min hourglass: 9 minutes left
Elapsed: 18 min (11 min measured)
At 21 minutes, the 7-minute hourglass runs out. Flip it.
7-min hourglass: 7 minutes left
9-min hourglass: 6 minutes left
Elapsed: 21 min (14 min measured)
At 27 minutes, the 9-minute hourglass runs out. Stop — 20 minutes measured.
7-min hourglass: 1 minute left
9-min hourglass: done
Elapsed: 27 min (20 min measured)
To visualise how this works, imagine flipping both 7-minute and 9-minute timers at the same time, flipping each one as it runs out.
- 7-minute hourglass finishes at: 7, 14, 21, 28...
- 9-minute hourglass finishes at: 9, 18, 27...
We want to start measuring at the 7 minute mark, and stop measuring at the 27 minute mark, giving us the 20 minutes we need. ::::
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