We can view braids mathematically as n strings hanging from a horizontal bar. Each piece of string runs downward and can cross neighboring strings. In the 1920s Emil Artin observed that braids of n strings form an algebraic group. To “multiply” two braids, we append the bottom of one braid with the top of another braid. The identity element in this group…

## A Numerical Crossword Puzzle

I recently made my first crossword puzzle. It was great fun. It had some mathematical clues, but it was not mathematical. So for my second crossword puzzle, I decided to make one that was 100% mathematical. Download my numerical crossword puzzle in which each cell contains a decimal point ” . ” or a digit 0…

## The Most Imaginary Number… is Real!

On the eve of “π Day,” a hush has fallen on the mathematical internet. Everyone is gearing up, getting ready to celebrate the beauty of the transcendental number π. Indeed, π is an amazing number. I have a book coming out in October about the geometric problems of antiquity (Tales of Impossibility: The 2000-Year Quest…

## My First Crossword Puzzle

I recently discovered Phil, an HTML5 crossword puzzle maker (here it is on GitHub). I’ve always wanted to make a crossword puzzle, but it seemed overwhelming. But with Phil, I found it fun and addicting! So, here is my first creation. It has a mathematical theme. I’ve enjoyed solving crossword puzzles off and on over…

## Make Your Own Pythagorean Cup

My parents recently went to Greece. They brought me back a souvenir—a practical joke cup called a “Pythagorean cup.” The legend behind the cup is that Pythagoras or one of the Pythagoreans invented this cup to prevent gluttony. The vessel looks like a cup with an odd pillar in the center. When you fill it…

## A bad, but interesting, exam question

Every teacher has had the experience of writing a seemingly straightforward exam question only to realize when grading the exam that some of the students misunderstood the intent of the question. Oh, to be able to turn back the clock and to rewrite the question! That happened to me this semester, and the variety of answers…

## Proof by Induction with a Difficult Base Case

I’m teaching Discrete Mathematics this semester. It is our “intro to proofs” class. One of the proof techniques the students learn is proof by induction. I told the class that usually the base case for induction proofs are easy and that most of the work occurs in the inductive step. Indeed, most of the proofs…

## Proof Without Words: Gregory’s Theorem

Archimedes famously used inscribed and circumscribed polygons to approximate the circumference of a circle. He then repeatedly doubled the numbers of sides to get an approximation for π. In 1667, James Gregory did the same, but he used areas: He discovered the following beautiful double-recurrence relation that can be used to compute the areas of inscribed…

## Möbius Band Ambigram

Almost 10 years ago I had some fun making ambigrams (a word or words that have some sort of symmetry—rotation, reflection, etc.) of my name. I posted some examples on this blog. Möbius bands have the surprising properties that they are one-sided and have only one edge. This inspired me to write the words MÖBIUS…

## Rubik’s Cube Tri-Hexaflexagon

A few days ago I came across an animated gif of a Rubik’s Cube hexaflexagon kaleidocycle that somone made. I posted it on Twitter. I've got to make a Rubik's Cube hexaflexagon! pic.twitter.com/ZxdSkby1iv — Dave Richeson (@divbyzero) August 21, 2018 It got a lot of interest, so I thought I’d try making my own. Here’s the final…