Ponder This Challenge:
Puzzle for January 2006.
Consider a loop of string of unit length. Suppose we cut the string independently and at random in n places. This will divide the loop into n pieces.
This month's puzzle asks
1. What is the expected (average) size of the smallest piece?
2. What is the expected (average) size of the largest piece?
If you can't find an exact answer the asymptotic behavior (to leading order) as n goes to infinity will suffice. Please give some sort of argument to justify your answer although we will not require it to be completely rigorous.
大意是:
假设有一单位长的绳圈。假设我们在n处独立的并随机的切割此绳圈。这就会将这个绳圈分割为n段。
问:
1)预期的(平均)最小尺寸是多少?
2)预期的(平均)最大尺寸是多少?
英文好的朋友在斟酌一下!
Puzzle for January 2006.
Consider a loop of string of unit length. Suppose we cut the string independently and at random in n places. This will divide the loop into n pieces.
This month's puzzle asks
1. What is the expected (average) size of the smallest piece?
2. What is the expected (average) size of the largest piece?
If you can't find an exact answer the asymptotic behavior (to leading order) as n goes to infinity will suffice. Please give some sort of argument to justify your answer although we will not require it to be completely rigorous.
大意是:
假设有一单位长的绳圈。假设我们在n处独立的并随机的切割此绳圈。这就会将这个绳圈分割为n段。
问:
1)预期的(平均)最小尺寸是多少?
2)预期的(平均)最大尺寸是多少?
英文好的朋友在斟酌一下!
