Author Archives: mzspivey

Intuition for the Dual in Linear Programming

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 | Leave a comment

The Sum of Cubes is the Square of the Sum

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

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Happy Birthday, Benoit Mandelbrot

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

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

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Some Video Learning Suggestions

During this COVID-tide many of us have been seeking out online learning resources.  I’ve done so quite a bit in the past few months, and I thought I would do a post to recommend some of these.  They are all … Continue reading

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A Lesson on Converting Between Different Bases

We’re in the time of COVID-19, and that has meant taking far more direct responsibility for my children’s learning than I ever have before.  It’s been a lot of work, but it’s also been fun.  In fact, I’ve been surprised … Continue reading

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A Coin-Flipping Problem

One problem that I’ve assigned when discussing Markov chains is to calculate the expected number of flips required for a particular pattern to appear.  (Here I mean a pattern such as heads, heads, heads, or HHH.)  In this post I’m … Continue reading

Posted in coin flipping, Markov chains, probability | Leave a comment

A Request for a Proof of a Binomial Identity

A few weeks ago I received an email from Professor Steve Drekic at the University of Waterloo. He asked if I knew of a way to prove the following binomial identity: (He told me later that he wanted it to … Continue reading

Posted in binomial coefficients, generating functions | 2 Comments

Strong Induction Wasn’t Needed After All

Lately when I’ve taught the second principle of mathematical induction – also called “strong induction” – I’ve used the following example to illustrate why we need it. Prove that you can make any amount of postage of 12 cents or … Continue reading

Posted in number theory, proof techniques | Leave a comment

Arguments for 0.9999… Being Equal to 1

Recently I tried to explain to my 11-year-old son why 0.9999… equals 1.  The standard arguments for (at least the ones I’ve seen) assume more math background than he has.  So I tried another couple of arguments, and they seemed … Continue reading

Posted in arithmetic, number theory | Leave a comment