Yesterday, I was made aware of an elementary combinatorics problem, and the solution was surprisingly clever (at least it is to me, though I am not an expert in the area).
Suppose that a philanthropist decides to give a total of ten dollars in one-dollar bills to 3 people. How many ways can we distribute the money, assuming that we don’t have to do so evenly?
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I saw this post on Reddit and was quite interested in it. I decided to investigate things on my own for a bit. We’ll start with the statement of the Cauchy Condensation Test.
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This post is a rewritten (and slightly extended) form of a survey paper that I wrote in 2014 for a seminar as a graduate student. Proofs for many of the results have been omitted as they are available in their respective papers, listed in the references at the end of this post. I will, however, list proofs that make important points (such as the proof of the Sierpiński-Mazurkiewicz Paradox which doesn’t use the Axiom of Choice) or were not given in the original source (as is the case for a lemma in the first paper of Sherman).
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A well-known fallacy committed by students is the so-called “Law of Universal Linearity” (the link is to a discussion of this phenomenon on Mathematics Stack Exchange). The most famous example of this is the statement
$$\left(x+y\right)^n = x^n + y^n,$$
known as the Freshman’s dream.
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I ask that anyone who has had multivariable calculus please stop right now and try to work out this problem.
$$\int_0^1\int_0^1\frac{dx\,dy}{1-xy}.$$
Hint: The integrand looks familiar!
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I have just finished moving all of my posts from the old location of this blog from a free webhost. I hope to be more productive from now on!
So as I study for my comprehensive exams, I will post some of my favorite problems from my studies.
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There are many rules for determining whether a number is divisible by another. For example, we know that a number is even (divisible by 2) whenever the one’s digit is even. We also know that a number is divisible by 3 if the sum of its digits is divisible by 3. A less known one is the rule for divisibility by 11, which says that a number is divisible by 11 when the number obtained by starting with the one’s digit, subtracting the 10’s digit, adding the 100’s digit, and so on, alternating between adding and subtracting, until you’ve used all digits, is itself divisible by 11. For example, we know that 628474 is divisible by 11 because $4-7+4-8+2-6=-11$, which is divisible by 11. Indeed, the quotient of 628474 by 11 is 57134.
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When I was in grade school learning about primes, I would ask myself: How many primes are there? If I pick a number at random, will it be prime?
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