Author Archives: mzspivey

Mathematics in Games, Part 1

When my wife and I were buying our first house many years ago our real estate agent Rita quickly found out that I was a math professor.  She gave a response I have heard frequently over the years: “I’m not … Continue reading

Posted in games, teaching | Leave a comment

Proof of the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states the following: Every integer greater than 1 can be represented uniquely as the product of prime numbers. Another way to put this is that every integer has a unique factorization.  For example, 60 factors … Continue reading

Posted in arithmetic | Leave a comment

Equivalence of Two Binomial Sums

This month’s post entails proving the following equivalence: Identity 1: Despite the fact that these two sums are over the same range and are equal for all values of n they are not equal term-by-term. If you think about the two sides … Continue reading

Posted in binomial coefficients, probability | Leave a comment

A Sum of Ratios of Binomial Coefficients

In this post we evaluate the sum .  Then we’ll generalize it and evaluate . The key tools we need for the first sum are the trinomial revision identity, , and the parallel summation identity, .  Using trinomial revision, we have . … Continue reading

Posted in binomial coefficients | Leave a comment

An Alternating Convolution Identity via Sign-Reversing Involutions

This month’s post is on the combinatorial proof technique of sign-reversing involutions.  It’s a really clever idea that can often be applied to identities that feature alternating sums.  We’ll illustrate the technique on the following identity. . (Here, we use … Continue reading

Posted in binomial coefficients, combinatorics | Leave a comment

A Mathematical Riddle

This past weekend I attended the wedding of a former student, Jake Linenthal.  On the dinner tables at the reception were sheets of paper containing a Mad Libs-style story of how Jake and his wife Abby got engaged, as well … Continue reading

Posted in arithmetic, puzzles | Leave a comment

Infinite Series Expressions for Pi, E, Phi, and Gamma

Recently I was trying to find infinite series expressions for some famous mathematical constants, and I thought I would record what I found here.  This post considers , Euler’s constant e, the golden ratio , and the Euler-Mascheroni constant . … Continue reading

Posted in sequences and series | Leave a comment