In my Discrete Mathematics class we discussed a few famous paradoxes, such as Russell’s paradox/barber paradox/librarian paradox, the liar’s paradox, and the naming numbers paradox. Afterward, a student of mine shared with me this old legal paradox featuring Euathlus and Protagoras. Euathlus wanted to become a lawyer but could not pay Protagoras. Protagoras agreed to…

# Month: November 2008

## Thoughts on how to teach induction

In their article “Some observations on teaching induction,” (MAA Focus, May/June 2008, pp. 9–10) Mary Flahive and John Lee give tips on how to teach induction. For a variety of reasons, they encourage professors to downplay proofs of theorems such as the “baby Gauss” formula for all . Indeed, I have noticed that students can…

## The prime number theorem in Calculus II

I attended Shahriar Shahriari’s MAA Minicourse Beyond Formulas and Algorithms: Teaching a Conceptual/thematics Single Variable Calculus Course at the 2008 Joint Mathematics Meeting. He talked about having his calculus students derive the prime number theorem. Recall that the prime number theorem states that if is the number of primes less than or equal to ,…

## Graph theory is funny

I read and laugh at Jessica Hagy’s Indexed every day. Today Jessica created the following index card titled 7 wonders of the modern world which is based on , the complete graph on 7 vertices. She has created similar index cards before, such as… : For my MBA friends (apologies to Mr. Porter) : Yes,…

## What are p-adic numbers?

I am not a number theorist, but I’ve always had a distant fascination with p-adic numbers. I have a list of “neat math topics” that I want to write about on my blog, and the p-adic numbers are on that list. So I was happy to see an interesting article about them by Andrew Rich…

## Jenga mathematics

I like this: on his blog The Endeavor, John D. Cook draws an analogy between strengthening a theorem and the game Jenga. Jenga is a game where you start with a tower of wooden pegs and take turns removing pegs until someone makes the tower collapse… I use the phrase “Jenga mathematics” to refer to generalizing…

## Do you give partial credit? How to grade Venn diagrams

Suppose that on an exam you asked your class to shade the region corresponding to in the figure below. The problem is worth 5 points. The correct answer is: When you received their solutions, some students had regions shaded that shouldn’t be shaded and left regions unshaded when they should be shaded. My question is:…

## Weapons of math instruction

Google is digitizing 10 million photos from the the LIFE Magazine archives. Search millions of photographs from the LIFE photo archive, stretching from the 1750s to today. Most were never published and are now available for the first time through the joint work of LIFE and Google. Here are a couple mathematical ones. Caption: Cadets…

## A correlation: 19th century cotton production and Obama votes

The Strange Maps blog has the following interesting map mash-up. This is an overlay of two maps. One is the 2008 presidential election results and the other is cotton production in 1860 (each dot represents 2000 bales). Strange Maps writes: The link between these two maps is not causal, but correlational, and the correlation is…

## Four shuffles suffice

It takes a while to shuffle a deck of cards seven times, but it is well known that that is how many riffle shuffles it takes to fully randomize a deck of 52 standard playing cards. This was shown in 1992 by Bayer and Diaconis. As it turns out, however, for some games fewer than…

## Judges, sonar, and innumeracy

Good Math, Bad Math has an interesting blog post about the recent Supreme Court case over the Navy’s use of sonar near marine wildlife. In the blog (and in the readers’ comments that follow), they talk about Chief Justice John Roberts use (misuse) of mathematics in his written decision.

## The sum of kth powers

Everyone loves the “baby Gauss story” in which Gauss amazes his teacher by quickly summing the first 100 positive integers in a flash of brilliance—he adds the first to the 100th, the second to the 99th, and so on to get the sum of fifty 101s to obtain 5050. (Brian Hayes has a great article…