Hourglass Problem, solution.

For a perfect breakfast, a fellow decides to boil an egg exactly 15 minutes. He has two hourglasses - one for 7 minutes, another for 11. How should he go about preparing his breakfast? How many times will he have to turn hourglasses? What would be the minimal required number of glass turns?

This is a clear case where the formal knowledge alone does not lead to a very good solution. To prove this point I'll give four solutions.

Solution #1

Let us again make use of the identity

a·X - b·Y = 1

Let a = 11 and b = 7. It's immediately verified that the identity is satisfied by X = 2 and Y = 3. This suggests the following strategy:

Start both glasses. When they become successively empty, turn the small glass, then the big one and the small one again. On its third run the small glass will empty 21 minutes from the start of the experiment. Start boiling your eggs at this point but do not yet turn the 7 min glass. After 1 minute (22 minutes after the beginning of the experiment) the big hourglass will become empty. Now, use the small glass twice to count 14 minutes. 1 + 14 = 15. In all, it will take 4 turns for the small glass and 1 turn for the bigger one.

Five turns is not too much. However, a shorter solution exists.

Solution #2 (Credits go to my wife Lana)

Start both hourglasses. At the end of 7 minutes restart the small one. At the end of 11 minutes restart the big hourglass. The small one at this point has 3 minutes to go. When it empties begin boiling your eggs. The big hourglass will run for another 8 minutes. After that use the small hourglass to measure another 7 minutes. 8 + 7 = 15. This solution requires turning the small hourglass twice and the big one just once which sums up to 3 turns.

Solution #3 (Credits go to my son David)

Again, start both glasses. After it empties, turn the small glass, start boiling your eggs and continue watching the big glass. When, at the end of 11 minutes, the big one empties the small glass will be running for 4 minutes. Turn the big glass and wait another 11 minutes. 4 + 11 = 15. All it takes is 2 turns of the big glass. It's an absolute minimum.

Remark

Thanks to Andre Gustavo dos Santos from Brasil for pointing out that turning the small glass is quite redundant.

Remark

In the spirit of the Three Glass Problem I'd like to note that Solution #3 is suggestive of the following formal statement:

Statement

Let there be two hourglasses with capacities a and b (b > a) minutes. Using these two glasses, is it possible to measure an interval c = 2b - a?

The answer is, of course, YES. Indeed, start both glasses. The moment the small glass becomes empty check your clock. From this point, it will take (b-a) minutes until the big one becomes empty. When it does, turn it and wait another b minutes. (b - a) + b = 2b - a.

Thus, starting with one problem we actually solved many others:

  1. For a perfect breakfast, a fellow decides to boil an egg exactly 13 minutes. He has two hourglasses - one for 5 minutes, another for 9. How should he go about preparing his breakfast? How many times will he have to turn hourglasses? What would be the minimal required number of glass turns?

  2. For a perfect breakfast, a fellow decides to boil an egg exactly 16 minutes. He has two hourglasses - one for 8 minutes, another for 12. How should he go about preparing his breakfast? How many times will he have to turn hourglasses? What would be the minimal required number of glass turns?

  3. And so on. (As was pointed out by Clive Townsend, the last problem has a trivial solution: using the 8 minutes glass twice. So, it's always nice to have a general solution, but such a solution is not necessarily the best possible.)

Solution #4 (Thanks to John Flynn)

Hi

I wanted to add my own friendly two-cent solution to the 15min egg problem.

At time0 start both timers, and begin to cook the egg.
At time7, timer7 expires- flip it over.
At time11, timer11 expires. Timer7 now has 3min of sand on top and 4min of sand on bottom. So flip over timer7!
At time15, timer7 expires. 15 minutes have passed.

Total of 2 flips. I agree that this doesn't (and can't) beat the one-flip solution, but it does allow the egg to be cooked immediately - not wait 7 minutes as Solution #3 requires. This of course is a big advantage given the cost of energy, and the fact that our need to purchase oil from other nations is a leading contributor to our nation's trade deficit problem!

However, given that this is the age of electronics, my solutions's drawback is that it TRULY requires 'hour-glasses'. That is, chronographs would not support my solution since it requires the device to count up as well as down at the same time. (the bottom sand is counting from 0 to 7 min, while the top sand is counting down from 7min to 0) Unless, the chronograph had both an up and down function, as many of today's Sport watches have; this is because sports clocks count time-remaining, not time-elapsed.

But of course, if you were using a watch, or equivalent, you would simply ime 15 minutes directly! But that wouldn't make much a puzzle now either.

Hope you enjoyed my light-hearted and long-winded resolution.

John

References

  1. M. Gardner, Mathematical Circus, Vintage Books, NY, 1981
  2. D. Wells, The Penguin Book of Curious and Interesting Puzzles, Penguin Books, 1992

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