# Making Progress on Beavers, AI, and Math

by Jonathan Kujawa Nearly two years ago we talked at 3QD about the Busy Beaver problem [1]. Since then, the beavers have been busy. As discussed in that essay, the Busy Beaver problem measures how complicated a computation might be. It does so by measuring how long a Turing might run before stopping. A Turing machine…

# The Hidden World of Gauss and His Periods

by Jonathan Kujawa One of the great pleasures in life is learning about something today that you couldn’t have imagined yesterday. The infinite richness of mathematics means I get to have this experience regularly. However much I think I know, it is a drop in the ocean of things yet to be learned. And even…

# Puzzles, Spherical Cows, and Applied Geometry

by Jonathan Kujawa My nieces, Hannah and Sydney, came to visit for the weekend. Since they’re 8 and 9, the delightful Guardian Games in downtown Corvallis was a must-stop. Along with games galore, they have an amazing assortment of puzzles. They have puzzles with micro-sized pieces, puzzles with jumbo-sized pieces, puzzles with only a few…

# Big Money Guaranteed!

# What are the odds?

by Jonathan Kujawa In 2016, here and here at 3QD, we talked about some of the inherent paradoxes in democratic voting [1]. We discussed Arrow’s Impossibility Theorem, along with related results like the Gibbard-Satterthwaite Theorem. They tell us that there is no way to convert the individual preferences of the voters into a single group…

# A Murmuration of Zeros

by Jonathan Kujawa Functions are machines that take in a number as an input, apply some rule to that input, and generate an output. The input might be cost, temperature, wind speed, a politician’s favorability rating, or whatever you like. The output could tell you the resulting profit, windchill, chance of a tornado forming, the…

# On the Typography of Numbers

by Jonathan Kujawa Mathematicians can be extraordinarily fussy about how they write. From having a near fetish on their choice of chalk to Donald Knuth taking 10 years in the middle of writing a multi-volume book series to develop an entirely new typesetting system [1], they spend an inordinate time thinking about how to write something.…

# A Spectre is Haunting Mathematics

by Jonathan Kujawa In March David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss announced that they discovered an “Ein Stein”. The choice of name can only be described as a tour-de-force of PR: Ein Stein translates as One Stone but also evokes a certain physicist. The Ein Stein got wide play in…

# On the Importance of Community

by Jonathan Kujawa In the movies the mathematician is always a lone genius, possibly mad, and uninterested in socializing with other people. Or they are Jeff Goldblum — a category of its own. While it is true that doing mathematics involves a certain amount of thinking alone, I’ve frequently argued here at 3QD that math…

# The Joy of Abstraction

by Jonathan Kujawa On “The Joy of Abstraction” by Eugenia Cheng. Category theory has variously been called “abstract nonsense,” “diagram chasing,” or the “mathematics of mathematics.” Some mathematicians find it a useful language, some a crucial tool for developing insights and obtaining new results, and more than a few have no use for it at…

# Human Flourishing in the Age of Machines

by Jonathan Kujawa Recently some colleagues and I were out to lunch. It was our University’s “Dead Week.” This is the week before finals when students are in a last-minute rush to finish projects and study for exams, and faculty are planning how to wind up their courses and beginning to draft their final exams.…

# On Busy Beavers and the Limits of Computability

by Jonathan Kujawa Some years ago we discussed Brobdingnagian numbers. You can find the 3QD essay here. These are numbers that are so mind-bogglingly large they defy human imagination. Perhaps the most famous is Graham’s number. It’s so large I can’t write it down for you. It’s not that I refuse to write it down; I can’t.…

# Life, the Universe, and Everything

by Jonathan Kujawa In May of 2020 we lost John Conway [0]. We discussed some of his mathematical accomplishments here at 3QD. He was a true original. At the time, I deliberately avoided discussing Conway’s most famous work: the Game of Life. Like a 60s rock band, Conway had mixed feelings about his most famous…

# Memories and Loss

by Jonathan Kujawa At the end of April heartbreaking news spread through my area of mathematics. Georgia Benkart had unexpectedly passed away. I first met Georgia when I was a graduate student, we’d seen each other at numerous conferences and other events, shared a few dinners, and I considered her a friend. Still, I was a little…

# Happy Birthday, King Friday XIII

# Saying A Lot While Saying Nothing At All

by Jonathan Kujawa In 1969, the mathematician Hans Freudenthal proposed a seemingly unsolvable puzzle: Abbas says to Sally and Penelope: I have two whole numbers between 1 and 100 which are not equal and whose sum is not more than 100. Abbas tells the sum of these numbers to Sally only, and their product to…