Water clocks puzzle
Mar. 1st, 2006 01:10 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackPuzzle: I have two identical cylindrical lemonade bottles I punched small holes into the sides of, just above the bottom1. I find that when I fill them each with water the first empties in 225 seconds and the second more quickly in 75 seconds. Without any other form of measurement, how do I use them to time 25 seconds?
Q. That's just one of those hourglass/numerology puzzles. If one hourglass empties in time p minutes, and the second in time q, then calculate p-1 mod q, and you know pp-1=1 (mod q) so pp-1=1+nq. Run glass Q n times and glass P p-1 times; when the first has finished you have exactly one minute before the second does.
A. If you think that's easy, try it.
Q. It's a trick, isn't it.
A. Probably.
Q. Tilt one of the bottles...
A. If you like. They *are* cylindrical, so you *can* halve the amount of water by tilting them until the water only just covers the base, if you think it'll help.
Q. I could use a...
A. No other equipment is necessary. You can use additional vessels of unknown size if you want to.
Q. How quickly did you solve it?
A. Actually, I made the puzzle up myself last night.
Q. Are you sure your solution works?
A. I may be mistaken. In which case I apologise, and promise to feel very embarassed.
Q. Do I need to use much maths for this?
A. Some. Nothing not taught at A-level IIRC.
Q. Can I find a different solution?
A. I don't think so, but try; it might be better.
[-1] That's not a footnote, that means 'inverse of p'
[1] Really.
Q. That's just one of those hourglass/numerology puzzles. If one hourglass empties in time p minutes, and the second in time q, then calculate p-1 mod q, and you know pp-1=1 (mod q) so pp-1=1+nq. Run glass Q n times and glass P p-1 times; when the first has finished you have exactly one minute before the second does.
A. If you think that's easy, try it.
Q. It's a trick, isn't it.
A. Probably.
Q. Tilt one of the bottles...
A. If you like. They *are* cylindrical, so you *can* halve the amount of water by tilting them until the water only just covers the base, if you think it'll help.
Q. I could use a...
A. No other equipment is necessary. You can use additional vessels of unknown size if you want to.
Q. How quickly did you solve it?
A. Actually, I made the puzzle up myself last night.
Q. Are you sure your solution works?
A. I may be mistaken. In which case I apologise, and promise to feel very embarassed.
Q. Do I need to use much maths for this?
A. Some. Nothing not taught at A-level IIRC.
Q. Can I find a different solution?
A. I don't think so, but try; it might be better.
[-1] That's not a footnote, that means 'inverse of p'
[1] Really.
no subject
Date: 2006-03-01 02:19 pm (UTC)no subject
Date: 2006-03-01 02:24 pm (UTC)Trial and error will probably satisfy you that you couldn't solve this problem if you had two hourglasses that took 75s and 275s.
Admittedly, my expected answer involved integration and newton's second law, so someone who wasn't good at A-level maths probably wouldn't be able to do it even if they'd studied the things once; for that I apologise.
no subject
Date: 2006-03-01 02:28 pm (UTC)no subject
Date: 2006-03-01 02:32 pm (UTC)