Let’s continue our puzzles discussion today with another puzzle type - time measurement using an hourglass.
First understand what an hourglass is - it is a mechanical device used to measure the passage of time. It comprises of two glass bulbs connected vertically by a narrow neck that allows a regulated trickle of sand from the upper bulb to the lower one. The sand takes a fixed amount of time to fall from the upper bulb to the lower bulb. Hourglasses may be reused indefinitely by inverting the bulbs once the upper bulb is empty.
This is what it looks like:
A 10 minute hourglass will let us measure time in intervals of 10 mins. All the sand will flow from the upper bulb to the lower bulb in 10 mins. We can then flip the hourglass and now sand will start flowing again for the next 10 mins and so on. We cannot measure, say 12 mins using just a 10 minute hourglass. But we can measure more time intervals when we have two hourglasses of different times. Let’s see how.
Puzzle 1: A teacher of mathematics used an unconventional method to measure a 15-minute time limit for a test. He just used 7 and 11-minute hourglasses. During the whole time he turned the hourglasses only 3 times (turning both hourglasses at once counts as one flip).
Explain how the teacher measured 15 minutes.
Solution: We have a 7 min and an 11 min hourglass. So we can measure time in intervals of 7 min as well as in intervals of 11 mins. But how about this: if both hourglasses start together, at the end of 7 mins, we have 4 mins of sand leftover in the top bulb of the 11 min hourglass. So we can measure 4 mins of time.
Further, if we flip the 7 min hourglass at this time and let it flow for 4 mins, we will have 3 mins of sand leftover and hence can measure a 3 min time interval too and so on…
Now, let’s see how we will measure 15 mins using a 7 min and an 11 min hourglass.
Start both 7 min and 11 min hourglasses. After the top bulb of the 7 min hourglass is empty, flip it. At this time, we have 4 min worth of sand in the top bulb of the 11 min hourglass. When the top bulb of the 11 min hourglass is empty, the bottom bulb of 7 min hourglass has 4 mins worth of sand.
11 mins have passed till now.
Now simply flip the 7 min hourglass again and let it run for 4 mins.
This is how we measure 11 + 4 = 15 mins of time.
Let’s look at another one.
Puzzle 2: Having 2 hourglasses, a 7 minute one and a 4 minute one, how can you correctly time 9 minutes?
Solution: Now we need to measure 9 mins using two hourglasses of 7 min and 4 min. We need to measure a time interval of 9 mins.
Start both the hourglasses. After 4 mins pass, we have all sand of the top bulb of the 4 min hourglass in the lower bulb. Flip the 4 min hourglass. In the 7 min hourglass, there is 3 min worth of sand in the upper bulb.
After 3 mins, all sand from the 7 min hourglass will be in the lower bulb and 1 min worth of sand will be in the upper bulb of the 4 min hourglass.
This is when our 9 min interval will start.
The 1 min sand will flow out from the 4 min hourglass.
Flip the 4 min hourglass and let all sand flow out. This will take 4 mins.
Again, flip the 4 min hourglass and let all sand flow out. This will take another 4 mins.
In all, we have measured a 1 + 4 + 4 = 9 min interval.
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