Muddy Hats
Inelegant Explorations of Some Personal Problems
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Years back, I encountered the following problem in a book on Number Theory: Prove that the sum \(H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\) is never an integer after \(n > 1\). In other words, in the sequence \(1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, \dots\), there is only one integer, and that is the first number \(1\).
A famous problem in probability is this: Suppose you randomly1 break a stick in two places. You will get three smaller pieces of stick. What is the probability that you can form a triangle using these pieces? One interesting thing about this problem is that depending on how you visualize breaking the stick, you get different probabilities.