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Topic: Four beetles and a square. (Read 749 times) |
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redPEPPER
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Four beetles and a square.
« on: Jan 2nd, 2003, 11:52am » |
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There's this square whose sides are one unit long. At each corner of the square is a beetle. Let's call them John, Paul, George and Ringo. From this position, the beetles are moving, at a constant speed, in the exact direction of the next beetle at all times. John tries to reach Paul, Paul goes towards George, George points at Ringo, and Ringo tries to catch John.
What distance will each beetle cross before they reach each other? Or do they even reach each other?
We assume that the beetles are nondimensional points. Reaching each other means being superimposed.
There are interesting questions to add but they might give out the answer so I'll let you chew on this for a while. I'll be watching for blunders in the wording of the riddle and will correct it if needed.
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william wu
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Re: Four beetles and a square.
« Reply #1 on: Jan 2nd, 2003, 12:25pm » |
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This is listed in the hard section as "four heatseeking missiles", but no one has discussed it yet according to the search function.
Handwavy argument for the distance each beetle will cross before they reach each other (hidden):
One unit. John, Paul, George, and Ringo always outline the vertices of a square. This square rotates and shrinks until finally, the Beatles converge to the center of the square. Now if John is chasing Paul, note that a) the distance between John and Paul is initially 1, and b) Paul always moves in a direction orthogonal to John's direction -- that is, Paul is not responsible for eventually closing the distance between himself and John. Thus John travels 1 unit, and by symmetry so does everyone else.
Admittedly I don't remember how to analyze this problem rigorously, it's been too long ...
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| « Last Edit: Jan 2nd, 2003, 12:26pm by william wu » |
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redPEPPER
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Posts: 160
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Re: Four beetles and a square.
« Reply #2 on: Jan 2nd, 2003, 1:46pm » |
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Oops! sorry for posting an already existing riddle. I read all riddles in easy, medium and microsoft, but I only chewed on a couple of the hard ones because... well, they're hard 
I don't know why this riddle would be in "hard" though. My criterion is: if I can solve it, it ain't hard  I even wondered if I shouldn't have posted this in "easy".
Anyway, as there's already a solution posted, I'll ask the additional questions I wanted to ask, which are related to the 0.999...=1 problem, which is the reason why I posted this in the first place.
Here goes:
- What is the shape of the path that each beetle will follow?
After you figure that out:
- How many times does each beetle spin around the center of the initial square before they meet? By now you should know that the beetles meet at the center, hehee
- assuming that the beetles move at a speed of one unit per second, how long will it take for them to meet? This one's not hard at all, but it relates to a point I made in the 0.999...=1 thread.
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ThudnBlunder
wu::riddles Moderator
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Re: Four beetles and a square.
« Reply #3 on: May 5th, 2003, 4:58am » |
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Quote:| What is the shape of the path that each beetle will follow? |
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An equiangular (or logarithmic) spiral.
Quote:| How many times does each beetle spin around the center of the initial square before they meet? |
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An infinite number of times.
Quote:| assuming that the beetles move at a speed of one unit per second, how long will it take for them to meet? This one's not hard at all, but it relates to a point I made in the 0.999...= 1 thread. |
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0.9999999999999999999999999999999999999999999999999999999999999......... seconds.
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| « Last Edit: May 5th, 2003, 5:01am by ThudnBlunder » |
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