This year, Occupy Math is presenting ornaments made with a recursive fractal method. As this is the seventh collection, we provide links for the earlier sets as well, below the fold. This holiday season is coming as we try to wind down the pandemic and endure the climate emergency. Do what you can about these planetary disasters, but also remember that the duties of charity and kindness lie on you as strongly as they ever have.

# Uncategorized

# Announcing a book on problem factories

Writing a book is a big project and Occupy Math is very pleased to announce the first book arising from, well, Occupy Math. This book started at a conference for math educators that Occupy Math was invited to by his former student Andrew McEachern, now a professor at York University. Something that happens when you get math educators together is they tell you their favorite motivating problems, the problems that get students excited and interested. Andrew and I noticed that many of these problems fit into large families of problems. We coined the term Problem Factory for one of these store-houses of problems. Later in the post we have a list of the problem factories that have appeared in Occupy Math posts, to give you the flavor of things and tempt you to purchase the book. The book is Mathematical Problem Factories *Almost Endless Problem Generation* and it gives a number of problem factories, each illustrated by collections of problems drawn from the factory.

Some readers may find the subtitle “Almost Endless Problem Generation” to sound like what has been happening to all of us during the pandemic. Occupy Math’s editor laughed for several minutes after reading it. Do not worry, in a math book, “problems” mean “exercises” and having lots of exercises is a good thing. At least for math teachers. Never mind.

# What do mathematicians do all day? Part VI

Today’s post is another exposition of research, this time on a game that my colleague Joseph Alexander Brown created for teaching artificial intelligence. The game is about the foraging behavior of moose, very much simplified, and is a good example for demonstrating how to make a strategically interesting game. The basic idea is simple. We have two moose that can choose between three different fields where they might forage. The fields have plants in them that grow back after being eaten, fast at first and then slower as the plants get back to full size. Each morning, the moose each choose a field. If they choose different fields, then they get a score equal to the forage in that field. If they choose the same field, they trumpet and threaten and tear up the field a bit but get no forage. Sounds simple, but there are some subtleties.

# Factor Grids, a Math Activity

Today we introduce a new activity, *factor grids*. Suppose we want to place the numbers 6, 10, 14, 15, 21, and 35 in a 3×2 grid, as shown below. The goal is to get the highest score, where we get points for the largest common factors between any two adjacent grids, added up. One of the best scoring solutions — worth 32 points — is shown below. The grid on the right shows the scores for numbers in adjacent grids in red and then adds them up. Fifteen and thirty five, for example, have five as a greatest common factor.

An instance of this puzzle requires a grid size and a list of numbers. Occupy Math has written code to find all possible solutions and then mines that for the best solution. This is a “brute force” approach, but for the grids in this post, with sizes up to 4×3, it works perfectly well.

# Is Mathematics Invented or Discovered?

A critical part of the question of the status of mathematics as a discovery or an invention is the *fallacy of false duality*. By asking if mathematics was invented or discovered, we imply that these are the two possibilities are what is available. There is an unending debate on this topic. When a debate does not end, this is often because the debate started with a false premise — in this case that mathematics is either invented or discovered. The second starting point for this post is a wonderful article, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, that establishes that mathematics is naturally the language of physics, the laws of which are clearly discovered.

# Back-to-School, Covid Edition

Occupy Math, now being chair of his department, is heavily involved in the plans for his institution’s reopening. The universities in his area have been engaged in a race to the bottom to be welcoming to students. Being welcoming to students is a good thing, usually — but in this case we will be packing thousands of students into residence while they are still in the process of being vaccinated and have several weeks to go before full effectiveness. Given that first year university students are suddenly not under parental supervision and looking for a richer, more interesting life than they previously had, this is a recipe for disaster. We have actual evidence because a much smaller number of students were let into residence last fall — before the delta variant — and had a 300 person no-mask close-packed party which led to an outbreak. This sets the stage, and we will be looking at the K-12 situation as well in the rest of the post.

# Image of the Week #271

This week, we have a network of shapes generated with Newton’s method.

# The sickle cell condition and covering up police brutality

Today’s post takes up another difficult issue, use of a usually benign medical condition to excuse or cover up murder under color of authority by police. The sickle cell trait, used in this fashion, is also a biologically interesting condition in its own right. This post will go through the biology and then move on to evidence that the sickle cell trait was used as an excuse — in the form of a falsehood by a medical examiner — to justify the death of a Black man in police custody. This is structural racism at its worst. Rather than investigating honestly for the medical cause of a death, someone whose job is to do exactly that is instead searching for the reason that the cause of death was not police brutality or negligence.

# Image of the Week #265

A generalized Julia set with three complex parameters. And a lot of colors!

# Adversarial Grading Encourages Cheating

Occupy Math has recently become the chair of his department, which means that he must review and sign every academic misconduct finding filed by members of the department. He also gets copies of how the Dean’s office decided to resolve the accusations. During the pandemic, the amount of cheating has gone up considerably, but it turns out that is just the first part of the awfulness. Another duty of the chair is giving faculty a sympathetic ear when they are having a difficult time. The students who are trying to cheat are mostly amateurs — they get caught because the faculty have set traps, but also because they cheat in a terribly obvious fashion. This post may help cheating students kick up their game, but that is not its intended purpose. A silver lining of the pandemic is that it highlights things we need to fix. The explosion of cheating highlights that the adversarial system of determining grades, always toxic, is extra-toxic right now. Surviving a disaster is something human beings do well, but they accomplish it by cooperating, something that an adversarial student-instructor relationship interferes with. This post looks at a number of aspects of academic dishonesty during the pandemic.