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Math Help - Pentagon

  1. #1
    Senior Member Sampras's Avatar
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    Pentagon

    Suppose to each vertex of a pentagon, we assign a number  x_i with  s = \sum x_i >0 . If  x,y,z are numbers assigned to three successive vertices and if  y<0 , then replace  (x,y,z) with  (x+y, -y, y+z) . Repeat this as long as  y<0 . Does this algorithm always stop? If it does stop, how many steps are needed? If not, why doesn't it stop? Prove that it stops or does not stop.
    Last edited by Sampras; June 22nd 2009 at 09:04 AM.
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  2. #2
    Senior Member Sampras's Avatar
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    Can we post solutions if no one has answered it?
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Yes please post a solution to this nice problem! I thought about it a little bit but I couldn't figure it out.
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  4. #4
    Senior Member Sampras's Avatar
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    It always stops. You need to find a function of the five vertices that decreases when you perform the operation. One such function is  f(x_1, x_2, x_3, x_4, x_5) = \sum_{i=1}^{5} (x_{i}-x_{i+2})^{2}, \ x_6 = x_1, \ x_7 = x_2 . Suppose  x_4 < 0 . Then we have  f_{new}-f_{old} = 2sx_4 < 0 .
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