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 a start by Charlie)

We know how to get time intervals of 1/2, 3/4, 7/8, 15/16 etc hours. So I'll
take that as given. To add 1/32 on the end of Charlie's 3/8, the only thing
I can see to do is this:

Start the necessary clocks (4) to measure 15/16 hours. When they hit
7/8 (and still running), start the necessary clocks (5) for 31/32 hours.
At time 15/16 (when the first set runs out), start Charlie's 3 clocks.
Start counting time 1/2 hour later. Then Charlie's 3 clocks have 1/4+1/8
left to run. When they expire, the (5) clocks still have 1/32 left to run.
(They started 1/16 sooner, and run 3/32 longer). Total time to measure
13/32 hours is 59/32 hours

I can see how to extend this to any finite decimal expansion of 5/12 hours.
The problem is, each stretch of 0's in the decimal expansion will require
adding at least 1/2 hour (more like an hour) to the total time.

Posted by Brian Wainscott on 2003-09-10 21:20:41