You have an infinite amount of timers, each is an hour long (they do not have dials on telling you how long they've been going - they just beep when the time is up). You can set it to double speed at any time, but you cannot set it back to normal speed (eg if you set it to double speed at the start it will last 30 minutes.
Using each timer only once, is it possible to time exactly 25 minutes?
If it is, what is the smallest number of timers you need to do this, and the quickest time you can acheive it?
(In reply to
Tic Toc by Alain)
While it is true that any of your timers can be built in less than an hour, how do you guarantee they all become available at the correct moment?
For example, to time X minutes, it is not just enough to construct two timers of duration A and B where A+B=X, you also have to build them in such a way that one of them expires at the precise instant that the other begins. One thing you CANNOT do is pause the clocks -- so you can't build a timer then pause it until you want to use it.
I think several things are clear:
1 - It can't be done with a finite number of timers. As many have pointed out, 5/12 isn't a finite fraction in binary decimal, so an infinite number of timers must be used, and you have to use some kind of limit process.
2 - Constructing each of the intervals (60 minutes/2^k) for each k is straight forward.
3 - getting all these intervals lined up is going to be the hard part, if it can be done at all in finite time.
I think Brian Smith has the nicest solutions so far: it can't be done, but you can come infinitely close if you are willing to work at it (total time required is about 60 minutes times the number of terms in the binary expansion of 5/12, and I like his approach in "Closer and closer")
re: Tic Toc
