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        riddles >> medium >> Lemming Drownings
        
(Message started by: william wu on Jan 22^nd, 2003, 1:04pm)

Title: Lemming Drownings
Post by william wu on Jan 22^nd, 2003, 1:04pm

Somewhere in Northern Eurasia, a group of 20 lemmings is planning a special group suicide this year. Each of the lemmings will be placed in a random position along a thin, 100 meter long plank of wood which is floating in the sea. Each lemming is equally likely to be facing either end of the plank. At time t=0, all the lemmings walk forward at a slow speed of 1 meter per minute. If a lemming bumps into another lemming, the two both reverse directions. If a lemming falls off the plank, he drowns. What is the longest time that must elapse till all the lemmings have drowned?

(if you make a certain cunning observation, this puzzle becomes trivial)

Note: Storyline inspired by the old video game Lemmings. Lemmings are mouse-like rodents rumored to commit group suicides, even though that's not really true -- a old Disney film inspired this myth. The video game Lemmings milks this myth. At the beginning of a level, hordes of the little creatures pour in a from the ceiling, and start walking forward foolishly, despite what perils may lay ahead (falling boulders, spiked pits, lakes of molten lava, etc.). The objective is to save them by building bridges, digging tunnels, and doing other ingenious manipulations of the environment.

Title: Re: Lemming Drownings
Post by S. Owen on Jan 22^nd, 2003, 2:30pm

Very satisfying little puzzle, Mr. Wu. With a knowing wink, I will agree that [hide]I never could tell those darn lemmings apart either.[/hide] Indeed.

Title: Re: Lemming Drownings
Post by Speaker on Jan 23^rd, 2003, 12:26am

Well, I want to clarify a few things.
What is the interval between lemmings?

I know this doesn't answer the question, but if you can fit ten 10cm long lemmings per meter, and all the lemmings were placed on the last two meters of the plank facing the end, then the shortest time would be two minutes as the little rodents trot right into the deep.

The longest time that might pass is quite a bit more difficult, and finding the answer is, I am chagrined to admit, not helped by S. Owen's hidden statement. :-[

Title: Re: Lemming Drownings
Post by william wu on Jan 23^rd, 2003, 1:13am

The lemmings are placed at random positions along the plank. I guess I'm implying equally random positions. So the interval between lemmings varies randomly. Treat the lemmings as dots with no thickness like in geometry, and the plank as a line.

Hehe. Some more hints:

[hide]
Forgot about the storyline -- just picture dots moving on a line in the manner described in the riddle. Perhaps then it will be eaiser to see how S.Owen's statement applies.
[/hide]

Title: Re: Lemming Drownings
Post by BNC on Jan 23^rd, 2003, 1:25am

A-ha! Would the answer be [hide]100 seconds[/hide]?

My reasoning is this:
[hide]
Say two lemmings, L1 and L2 walks toward each other. Say L1 walk in the +X direction, and L2 in the -X. Now they bump. L1 reverses direction to -X, and L2 to +X. So what? There is still one lemming going +x, and 1 lemming going -X. So, once one lemming starts walking toward the edge from distance D, one lemming will fall off (maybe a different one) after D/V (V being the velocity). So, longest time would be if a lemming was to be positioned at the very end of the plank.
[/hide]

Title: Re: Lemming Drownings
Post by william wu on Jan 23^rd, 2003, 2:29am

Yes, congratulations :) The velocity was defined as 1 meter per minute though, so the answer would be [hide]100 minutes[/hide]. Key observation is that (SPOILER) [hide]two lemmings bumping into each other is comparable to two lemmings passing each other.[/hide]

Title: Re: Lemming Drownings
Post by wowbagger on Jan 23^rd, 2003, 2:31am

I agree with your reasoning, BNC, but the correct answer is [hide]100 minuts[/hide], right?

S.Owen:
That was really nifty wink. :)

Title: Re: Lemming Drownings
Post by william wu on Jan 26^th, 2003, 4:49am

Decided to move this one to medium after running it by some more friends.

Title: Re: Lemming Drownings
Post by Speaker on Jan 26^th, 2003, 7:58pm

Okay, I say 100 minutes too, which I figured out before peeking. That is I peeked at S.Owens comment, but not at the others. But, I have looked at them now.

I used the same reasoning, that if two lemmings are walking towards each other, then they will need to cover the same distance whether they rebound off each other or if they pass through each other.

Anyway, late, but not forgotten.