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Which (if any) of these intervals is it possible to measure given two sand timers (which change from question to question)? There are no minute marks, halfway points, etc.

A. 7 minutes with 4- and 5-minute timers

B. 15 minutes with 7- and 11-minute timers

C. 14 minutes with 5- and 8-minute timers

If one (or more) is possible, specify how it can be done with the given timers.

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    $\begingroup$ In general, as long as the timers measure a relatively prime number of minutes, you can measure any number of minutes with the two timers. $\endgroup$
    – user88
    Aug 6, 2014 at 4:04
  • $\begingroup$ @JoeZ. What can be said about two number of minutes which are not relatively prime? Maybe that the number of minutes that you are measuring must then be a combination of the other two? $\endgroup$
    – Neil
    Aug 26, 2014 at 15:11
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    $\begingroup$ Yes. In general, if you have two timers of $x$ and $y$ minutes, any linear combination of the two ($ax+by$) of minutes can be measured. $\endgroup$
    – user88
    Aug 28, 2014 at 2:21

3 Answers 3

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A is possible: Turn both the 5 and the 4. When the 4 runs out, flip it over.Now, when the 5 runs out, start timing. The 4 will run for three more minutes, after which, you can flip it over to reach 7.

B is possible: Turn both the 7 and the 11. When the 7 runs out, start timing. The 11 will run for 4 more minutes, after which it can be flipped to reach 15.

C is possible: Turn both the 5 and the 8. When the 5 runs out, flip it. The 8 will then run out after 3 minutes, leaving 2 minutes in the 5. Flip the 8 then. When the 5 runs out, start timing. There are now 6 minutes left in the 8, and flipping the 8 after those 6 minutes gives 6 + 8 = 14 minutes.

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Another approach for A is: Flip both the 4 and the 5 and start timing. When the 4 runs out flip it. When the 5 runs out flip both, the 4 has now 1 minute left. When the 4 runs out flip both again, now the 5 has 1 minute left. When the 5 runs out, you have 7 minutes.

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You can do it even if you start timing when you first flip both timers. When either timer runs out, flip both timers.

A. When you flip the 4 minute timer, the 5 minute timer has 1 minute left. After that, you'll flip both timers at 5 minutes, 6 minutes, and 7 minutes.

B. When you flip at 7 minutes, the larger timer has 4 minutes left. Flip both again at 11 minutes, and both again at 15 minutes, and you are done.

C. When you flip both at 5 minutes, the larger timer has 3 minutes left. Flip both again at 8, then both again at 11, then both again at 14, and you are done.

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