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Four Pins and a Rubber Band (Posted on 2005-02-10) |
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Four pins are randomly nailed to a square board (according to a uniform probability distribution), and a rubber band is stretched completely around them to form a convex shape. What is the probability that the rubber band is in the shape of a triangle?
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Submitted by David Shin
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Rating: 4.1000 (10 votes)
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Solution:
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(Hide)
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The answer is 11/36. This is actually a special case of Sylvester's Four Point Problem, whose solution is outlined in the following website:
http://mathworld.wolfram.com/SylvestersFour-PointProblem.html
When I submitted this problem, I did so without actually
having computed an answer, though I figured it would not be overly complicated to compute. The above website indicates that the computation is an in fact an extremely difficult one; if I had known this previously, I would not have submitted the problem on perplexus. My apologies. |
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