Author Archives: mzspivey

Optimizing Against the Field: A Bracket Pool Strategy Using Team Seeds, Part 1

Posted on April 10, 2026 by mzspivey

This year’s NCAA basketball tournament has just ended, which means a lot of people are currently cursing their bracket picks. That also means it’s a great time for a retrospective: How could one go about picking a better bracket? There … Continue reading →

Posted in games, optimization, probability, sports, statistics | Leave a comment

Mean Residual Life for the Weibull Distribution

Posted on March 23, 2026 by mzspivey

A few years ago I answered a question on Cross Validated about the existence of a closed form for the mean residual life of the Weibull distribution, a popular distribution in reliability analysis. The short answer is that it depends … Continue reading →

Posted in probability, special functions, statistics | Leave a comment

An Interview Question about Car Replacements

Posted on February 26, 2022 by mzspivey

Recently I came across the following on a site that provides questions on probability and statistics for tech interviews. I solved it, and so I thought I would give my solution here. Suppose there is a new vehicle launch upcoming. … Continue reading →

Posted in interview questions, probability | 1 Comment

Two Fallacious Induction Arguments

Posted on September 16, 2021 by mzspivey

In this post I’m going to discuss two “proofs” by induction. That is, they’ll both be attempts at proving a statement, but they’ll both have flaws in their arguments. My hope is to alert people to potential errors when attempting … Continue reading →

Posted in proof techniques | 1 Comment

Some Corrections to My Paper “A Combinatorial View of Sums of Powers”

Posted on July 9, 2021 by mzspivey

A few months ago Mathematics Magazine published a paper of mine, “A Combinatorial View of Sums of Powers.” In it I give a combinatorial interpretation for the power sum , together with combinatorial proofs of two formulas for this power … Continue reading →

Posted in combinatorics, number theory, publishing, Stirling numbers | Leave a comment

A Connection Between the Riemann Zeta Function and the Power Sum

Posted on June 18, 2021 by mzspivey

The Riemann zeta function can be expressed as , for complex numbers s whose real part is greater than 1. By analytic continuation, can be extended to all complex numbers except where . The power sum is given by . … Continue reading →

Posted in Bernoulli numbers, number theory | Leave a comment

Intuition for the Dual in Linear Programming

Posted on March 31, 2021 by mzspivey

One of the most important theoretical results in linear programming is that every LP has a corresponding dual program. Where, exactly, this dual comes from can often seem mysterious. Several years ago I answered a question on a couple of … Continue reading →

Posted in linear programming, optimization | 1 Comment

The Sum of Cubes is the Square of the Sum

Posted on December 17, 2020 by mzspivey

It’s fairly well-known, to those who know it, that . In other words, the square of the sum of the first n positive integers equals the sum of the cubes of the first n positive integers. It’s probably less well-known … Continue reading →

Posted in number theory | Leave a comment

Happy Birthday, Benoit Mandelbrot

Posted on November 20, 2020 by mzspivey

Today’s Google doodle honors mathematician Benoit Mandelbrot. He would have been 96 today. If you’re interested in learning more about his life and work, the Google doodle link contains a short summary. If you want to go deeper, you can … Continue reading →

Posted in complex numbers, fractals, people | Leave a comment

No Integer Solutions to a Mordell Equation

Posted on October 30, 2020 by mzspivey

Equations of the form are called Mordell equations.  In this post we’re going to prove that the equation has no integer solutions, using (with one exception) nothing more complicated than congruences. Theorem: There are no integer solutions to the equation … Continue reading →

Posted in number theory | Leave a comment