Too Easy, I'm Sure, But Fun

[CzarJ]

Here's a puzzle I stole shamelessly from a site which shall remain nameless until somebody figures it out (I changed the wording a little bit):

One hundred ants are dropped randomly (and in random places) on a meter stick, and start walking randomly either left or right at a speed of 1 cm/sec. When two ants meet on the stick, they reverse directions and keep walking. When they reach the end, they fall off (don't worry, they're fine). So, the question is: How long will you have to wait before you can guarantee that the stick is ant-free?

Oh, by the way, I'm starting a pool. I give it five minutes.

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Unslumping yourself is not easily done.

[mith]

I'm going to take a wild guess and say 100s.

[mith]

Reasoning: In the simple two ant case, the longest time is clearly 100s. There are several ways this can happen. One ant can start at the end and walk the entire length, while the other is ahead of it walking the same direction; one can start at the end and the other can start *anywhere* on the stick going the other direction, and that one will take 100s (for example, if one starts at the 0cm mark, and one starts at the 80cm mark, they will meet at the 40 and then the one that started at 80 will have 60 more to go).

In the 100 ant case, let one ant start at one end (call it A), and walk in the direction of the rest of the stick. Let the other ants fall however, and look at the ant closest to the other end that is headed toward A (if they are all walking in the same direction, it will obviously take 100s). Call this ant B. Now, B will travel until it hits a third ant, C (or, until it hits A, in which case we have gotten back to the two ant solution). C will begin walking in direction B was going, starting from the same spot, at the same time, and it is now the closest ant to the other end that is walking toward the A. So, C has taken the role of our B ant, while B is now headed freely toward the end of the stick. Now C will run into D (or A, in which case we are, again, done), and so on. So, it's basically the two ant case, with more craziness.

[mith]

In fact, if you know which ant is farthest away from the end it is walking toward, this distance/the rate is the time it will take.

[CzarJ]

Hmm... 22 minutes, not too bad.
The basic reasoning at the site I got the puzzle from was that if two ants hit each other and start traveling in opposite directions, it is in essence exactly the same as if they passed right by each other. So the obvious answer is 100 seconds indeed. Anyway, here's the site for those who are interested: http://www.math.hmc.edu/funfacts/ Lot of interesting stuff, at least the stuff I can understand (which is, surprisingly, most of it=P).

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Unslumping yourself is not easily done.

[CzarJ]

Hmm... that link was supposed to invisibilize. Oh well, doesn't really matter.

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Unslumping yourself is not easily done.

[groza528]

links don't work under the invisible tags for the same reason images don't. The invisible tags change the color of the text to match the color of the background. Links have their own color. Hy might be able to write this one in though.

[mathgrant]

Test. . .
The GL!!

To invisible a link, you must put the invisible tags INSIDE the link tags.

[This message has been edited by mathgrant (edited 03-02-2002 04:11 PM).]

[CzarJ]

Test...

[IMG]http://www.math.ukans.edu/~rgrondahl/calculus.gif[/IMG]

(edit: well... sorta)

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Unslumping yourself is not easily done.



[This message has been edited by CzarJ (edited 03-04-2002 06:28 PM).]

[HyToFry]

Excellent work mathgrant.

[Griffin]

Hmm...

[CzarJ]

Cheater

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Unslumping yourself is not easily done.