# A Sand Timer Challenge

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.

• 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. – Joe Z. Aug 6 '14 at 4:04
• @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? – Neil Aug 26 '14 at 15:11
• 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. – Joe Z. Aug 28 '14 at 2:21

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.

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.

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.