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Re:怀疑这个题的物理前提 是错的 我也这么认为,但我还有别的想法

Posted by angeldust at 2012-03-05 20:35:48 on Problem 1003 and last updated at 2012-03-05 20:47:52
In Reply To:Re:怀疑这个题的物理前提 是错的 我也这么认为,但我还有别的想法 Posted by:wanghexiang221 at 2006-10-27 08:45:47
> 
> 我觉得好象是
> 1/(2*1) +1/(2*2) +1/(2*3) +1/(2*4) +.....
> =(1/2)*(1/1+1/2+1/3+1/4+....)
> 
> 我的推导是这样的:
> 设Si为前i张扑克最大探出距离,S0=0,S1=1/2
> 由重心公式,加上第i+1张扑克后最大的探出距离是:
> Si+1 = ( Si*(i+1) + 1/2 ) / (i+1) , i=0,1,2,3,...
> Si+1 - Si = 1/(2*(i+1))

确实每次只应该延伸1/(2*n),证明如下:

设牌的长度为1,第n次牌延伸长度为x,一张牌的重量为m

那么重心点右边增加的重量为: 
n*m*x
左边增加的重量为:
(1-x)*m-(n-1)*m*x(上面n-1张牌都向右边移动了x,最下面一张牌有1-x的长度在左边)

于是算出x=1/(2*n).

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