The other day I wanted to figure out a way to calculate every time on the clock where the minute hand and the hour hand lie directly over one another. I think I've got it figured out but I'm not 100% sure. Let's think of the time as x:y, x being the hour and y being the minute. Even though seconds are not units of distance, we can measure the distance of one full rotation as 60 "segments." Now for every 12 that the minute hand moves, the hour hand moves 1, so the ratio is 12:1. For every full number the hour hand advances, it is actually 5 minutes or segments. So combining that fact with the 12:1 ratio I came up with the equation y = 5x + y/12 and to get only one y in the equation y - y/12 = 5x 12(y - y/12) = 12(5x) 12y - y = 60x (11y)/11 = (60x)/11 y = 60x/11 Now let's say we want to figure out the specific time between 1:00 and 2:00. The hour will be 1 so we'll plug in 1 to the equation y = 60(1)/11 y = 60/11 y = 5.45 So if x=1 and y=5.45 the time will be 1:05.45 with the decimal being a fraction of a minute. To turn that into seconds we'll just multiply the .45 by 60 to get 27.0 seconds. If my calculations are correct, the hour hand and the minute hand will lie directly over one another at approximately 1:05 and 27.3 seconds and the other times can all be calculated by substituting numbers 1-11 for x. The times in hours:minutes.seconds would be 12:00, 1:05.27.0, 2:10.54.6, 3:16.21.6, 4:21.49.2, 5:27.16.2, 6:32.43.8, 7:38.10.8, 8:43.38.4, 9:49.05.4, 10:54.33.0 To show something interesting, say we want to figure out what time this will happen at the hour 11. This will not happen because it will get to 12:00 but let's plug in 11 anyway y = (60 x 11)/11 y = 660/11 y = 60 This will leave us with the time 11:60, which is 12:00, so it all works out nicely.

Quite a good little question really. Much harder than you expect. A good thing to get people thinking is to do with Xeno's paradox. A person has to get from A to B, lets say 100m in an infinite amount of time. In order to do this he would need get to the mid point, 50m, then the midpoint of that, 75m etc. How long does it take to get from A to B?

That's a classic calc I question. You'll never arrive, because even though the distance may be infinitesimally small, you still will not have reached point B. Eskimo, that's way cool to be able to take a problem and translate it into a formula like you did.

So there becomes an infinite number of steps, but as the step size approaches 0, so does the time to traverse it. Then you basically wind up with an indeterminate fraction which you can take the limit of... blah blah blah... Mathematics resolves it, but philosophers don't believe it. That is why I hate philosophy.

Well, a person would get from point a to point b because a person would be incapable of moving in those infinitesimally minute steps and/or would end up getting frustrated and just walking over the line.

Kind of like the question, "How many licks does it take to get to the center of a tootsie roll pop?" The owl loses its patience and doesn't make it past 3 licks.

I once took an ancient greek philosophy class, and after my prof explained this idea, I raised my hand and said something like "personally, I think this is total BS." I'm not really sure why I did it, but it was the boldest thing I've ever done in any class. All I remember of his response was that he didn't disagree with me

Isn't this pretty much the same issue as radiation half-life? Every half-life the substance gives off half as much radiation, and it will *never* reach zero radiation, but the remaining radiation particles (alpha's? beta's? gamma's?) will take infinitely long to appear.

It will never reach zero if it weren't for atoms. For simplicity, the statistical average of a substance with say 16 radioactive atoms would be to decay down to half over every half-life. Hence you'd go 16->8->4->2->1. Then I guess you're stuck as you can't divide one in half, but then you have to realize that the half-life of a substance is a statistical construction based on the probability of a certain quantum event occurring. Then you realize that the particle will decay, just it is difficult or impossible to say precisely when it will happen.

sorry, eskimo. didn't make it past there. just reminded me too much of math competition questions, i guess. yeah, there's some interesting stuff you can do with just algebra under your belt. but the more math you know, the more problems like this you can tackle. for instance, after i read a calc book, i thought of this one problem, a proof for something. now... kinda like the problem that you did, nothing particularly practical, per se, came out of it. but not only was it a good start to my science/math education, i also got $400 for an essay i wrote describing my experience doing this. (yay for scholarships!)