IBM Ponder This #093 · January 2006

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.

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The first 100 people who answer correctly will be listed. The answer will be posted a week after the 100th is received, or at the end of the month.

Solution

Best opened after a real attempt

To be added.