Category Archives: logic

Ways to prove a bijection

Posted on March 17, 2020 by Brent

You have a function and want to prove it is a bijection. What can you do? By the book A bijection is defined as a function which is both one-to-one and onto. So prove that is one-to-one, and prove that … Continue reading →

Posted in logic, proof | Tagged , , , , , , , , | 7 Comments

One-sided inverses, surjections, and injections

Posted on March 14, 2020 by Brent

Several commenters correctly answered the question from my previous post: if we have a function and such that for every , then is not necessarily invertible. Here are a few counterexamples: Commenter Buddha Buck came up with probably the simplest … Continue reading →

Posted in logic | Tagged , , , , , , | 1 Comment

Test your intuition: bijections

Posted on March 3, 2020 by Brent

Suppose we have sets and and a function (that is, ’s domain is and its codomain is ). Suppose there is another function such that for every . Is necessarily a bijection? That is, does necessarily match up each element … Continue reading →

Posted in logic | Tagged , , , | 12 Comments

The wizard’s rational puzzle (solutions, part 2)

Posted on November 9, 2018 by Brent

At long last, here is the solution I had in mind for the Wizard’s rational puzzle. Recall that the goal is to figure out the numerator and denominator of a secret rational number, if all we are allowed to do … Continue reading →

Posted in arithmetic, challenges, logic, programming, puzzles, solutions | Tagged , , , , , , , , , , | Comments Off on The wizard’s rational puzzle (solutions, part 2)

The wizard’s rational puzzle (solutions, part 1)

Posted on October 24, 2018 by Brent

About two and a half months ago I posted a challenge involving a sadistic math wizard, metal cubes containing rational numbers, and a room full of strange machines. I’ve been remiss in following up with some solutions. (Go read the … Continue reading →

Posted in arithmetic, challenges, logic, programming, puzzles, solutions | Tagged , , , , , , , , | 3 Comments

The wizard’s rational puzzle (mind your p’s and q’s!)

Posted on August 7, 2018 by Brent

You have been abducted by a sadistic math wizard (don’t you hate it when that happens?). He ushers you into a plain but cozy-looking room, with a hardwood floor, a few exotic-looking rugs, and wood paneling on the walls. He … Continue reading →

Posted in arithmetic, challenges, logic, programming, puzzles | Tagged , , , , , | 15 Comments

Book review: Roads to Infinity

Posted on July 5, 2011 by Brent

What is infinity? What is proof? These are two of the biggest questions mathematicians have grappled with over the years. In this well-written and fascinating book, John Stillwell takes us on a tour through some of the answers to these … Continue reading →

Posted in arithmetic, books, computation, induction, infinity, logic, proof, review | Tagged , , , | Comments Off on Book review: Roads to Infinity

My blog is back (hooray!) and a quick puzzle

Posted on February 22, 2011 by Brent

My blog is finally back up after a long hiatus due to hosting problems (not the first time). I’m seriously contemplating moving somewhere more reliable, but it’s hard to come up with a solution that would not (a) invalidate any … Continue reading →

Posted in logic, meta, puzzles | Tagged , , , | 8 Comments

Manufactoria

Posted on May 25, 2010 by Brent

A friend of mine just pointed me to a most excellent puzzle game, Manufactoria, wherein you build little machines to test robots. For now I won’t give away the secret of what real math/computer science topic the game teaches you, … Continue reading →

Posted in games, links, logic, programming, puzzles | Tagged , , | 17 Comments

The haybaler

Posted on December 16, 2009 by Brent

At Penn Alexander’s math club yesterday, the students worked on a fun puzzle that I’d never seen before. It goes like this: You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in … Continue reading →

Posted in arithmetic, challenges, logic, puzzles, teaching | Tagged , , | 11 Comments