Three Circles in a Triangle: A Solution

In my last post, I posed the problem of showing that if triangles $ABC$ and $PQR$ (as shown in the diagram below) are similar, then $\Delta ABC$ is equilateral. We will solve this problem in two steps: Show that the… Continue Reading

Three Circles in a Triangle: A Problem

This week, I was playing around with some geometry and came up with a fun problem. Suppose there is a triangle, $\Delta ABC$. Inside the triangle are three circles, each of which is tangent to the other circles and to… Continue Reading

Math Kangaroo

The movie Mean Girls took some liberties with its Mathletes scene, but it mostly got the essentials of math competitions right. In a typical competition, a handful of students from each school face off to solve some hard problems. I started… Continue Reading

Maximizing the Product: An Explanation

In my last two posts, I asked what the maximum product of integers adding up to 2017 was and then showed that the answer was $2^2\cdot 3^{671}$. It was not clear at the beginning that the largest possible product would… Continue Reading

Maximizing the Product: A Solution

In my last post, I asked What is the largest possible product you can make with a group of positive integers if the sum of those integers is 2017? Before we rush to the numerical answer, let’s follow my usual… Continue Reading

Maximizing the Product: A Problem

This week I offer a problem: What is the largest possible product you can make with a group of positive integers if the sum of those integers is 2017? Here we do not assume that the integers are distinct from… Continue Reading

My last post put forward the challenge to find numbers whose binary representations end with their decimal representations, like how $11 = 1011_2.$ It turns out that there are lots of them. Infinity of them, in fact. If we’re going to… Continue Reading

Binary Patterns: A Puzzle

Imagine a world in which there are only 2 digits. We are used to having 10 digits, from 0 to 9, but with just 2 of them, we’d be limited to 0 and 1. We could count in that world… Continue Reading

Infinity

Generally speaking, people are not indifferent to math. When I introduce myself as a mathematician, I rarely get the disinterested “oh”s I’d expect if I were instead middle management at an unfamiliar, generically-named firm. Sometimes I get math horror stories,… Continue Reading

A Mathematician’s First Camping Trip

My son has reached the age where being a scout means being outdoorsy. Last weekend, he and I went on our first camping excursion with his den. Scouting urges to BE PREPARED, and I didn’t want to be the one… Continue Reading