Author Archives: Brian
Some Fun with Combinatorics: How many ways can you give 10 one-dollar bills to 3 people?
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 … Continue reading
On Sums of Reciprocals with Logarithmic Factors (or, The Generalized $p$-Series Test)
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.
Paradoxical Decompositions in the Plane
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 … Continue reading
Freshman’s Dream Come True! (But only in characteristic $p$)
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 $$\left(x+y\right)^n = x^n + y^n,$$ … Continue reading
A Fun Integral Problem
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!
Site Transfer Complete!
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!
Studying for Comprehensive Exams (Part 1)
So as I study for my comprehensive exams, I will post some of my favorite problems from my studies.
Divisibility Rules: An Introduction to Modular Arithmetic
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 … Continue reading
Primes and Probabilities
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?