### Author Topic: 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)  (Read 42100 times)

#### 七把叉

• Jr. Member
• Posts: 27
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #15 on: 四月 01, 2005, 05:43:58 pm »
Quote from: 七把叉

#### fzy

• Hero Member
• Posts: 520
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #16 on: 四月 01, 2005, 06:06:17 pm »
Quote

The proof by contradiction can be fixed. But you have not fixed it yet. It is actually very easy to fix: Instead considering the maximum extend of all constructions, consider the maximum entends of all monotone constructions. No changes for the rest.

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It looke like is. But your proof does not look good enough. You proved it is "locally optimal", but have not proved it is globally optimal.   [/quote]

#### 七把叉

• Jr. Member
• Posts: 27
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #17 on: 四月 01, 2005, 06:15:28 pm »

#### 七把叉

• Jr. Member
• Posts: 27
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #18 on: 四月 01, 2005, 06:32:36 pm »
Quote from: fzy

The proof by contradiction can be fixed. But you have not fixed it yet. It is actually very easy to fix: Instead considering the maximum extend of all constructions, consider the maximum entends of all monotone constructions. No changes for the rest.

#### 差不多

• Newbie
• Posts: 10
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #19 on: 四月 01, 2005, 09:56:13 pm »

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#### warren

• Full Member
• Posts: 148
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #20 on: 四月 02, 2005, 01:10:49 am »
Quote from: 万精油
According to earlier posts, the nth brick will extend out 1/(2*n), which is getting smaller and smaller (the sum will go to infinite of course). By your new approach, if I split the top brick into two (thus, going from n brick to n+1 brick), I can always extend it 0.11 out. This is much bigger than 1/2*n, what happened?

Note: I know my above argument is wrong, I just want to see if people can find out where did I go wrong? (hint: physics)

That is because of the top 2 bricks are thiner and lighter than other bricks.

#### warren

• Full Member
• Posts: 148
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #21 on: 四月 02, 2005, 01:15:11 am »
Quote from: 差不多

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#### 差不多

• Newbie
• Posts: 10
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #22 on: 四月 02, 2005, 07:41:47 am »
Quote from: warren
Quote from: 差不多

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#### zzzzzzzzzz

• Guest
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #23 on: 四月 02, 2005, 01:07:12 pm »
Quote from: 万精油
According to earlier posts, the nth brick will extend out 1/(2*n), which is getting smaller and smaller (the sum will go to infinite of course). By your new approach, if I split the top brick into two (thus, going from n brick to n+1 brick), I can always extend it 0.11 out. This is much bigger than 1/2*n, what happened?

Note: I know my above argument is wrong, I just want to see if people can find out where did I go wrong? (hint: physics)

Not sure if I understood your question correctly, 1/2n applies to the bricks on the bottom.  The top brick can always extend out 0.5 > 0.11.

#### 七把叉

• Jr. Member
• Posts: 27
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #24 on: 四月 03, 2005, 09:49:14 am »
Quote from: 差不多

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#### physics

• Guest
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #25 on: 四月 07, 2005, 06:28:34 pm »
the method by z^10 gives the optimal result.
It can be showed in the following way.
If for any given n bricks, sum (1/2*n) is the best result, then ok.

Otherwise, suppose for some m,
the result can be greater than sum (1/2*m)
so there must exist some k<m, such that
for any numbers less than k,
the extension is not greater than sum (1/2*a), for 1<=a<=k

while for k+1 bricks, it is greater than sum 1/2*(k+1)

We know it is impossible. To show it is impossible.

For the (k+1)th brick
it should satisfy two conditions,
1st, its extension relative to the first brick is greater than sum(1/2*(k+1)),
2nd, its relative extension to the second brick should be not greater than sum(1/2*k),
but it is impossible according z^10's method of calculating center of mass.

#### 万精油

• Hero Member
• Posts: 1831
##### 每周一题: 砖块延伸 (3/28/05 -- 4/3/05)
« Reply #26 on: 四月 08, 2005, 11:15:17 pm »
Check out the following link, it can be used as the title picture of this thread.

http://web.wenxuecity.com/BBSView.php?SubID=joke&MsgID=91152