Math Digests December 2022

A brief history of statistics in football: why actual goals remain king in predicting who will win

The Conversation, December 30, 2022.

What kind of information is most helpful to predicting the outcome of a soccer game? There is lots of data you could try to incorporate in a prediction, but the best information comes from one of the humblest measurements: The number of goals historically scored by each team. Other kinds of data have been analyzed, like amount of time in possession of the ball, or number and quality of goal-scoring opportunities. But these don’t add much to predictions about who will win, writes Laurence Shaw in The Conversation. That’s because only scored goals have a clear and certain impact on the game’s outcome. This failure of flashy statistics reminds us of what really matters for mathematical prediction. “A model that only uses goals to predict future games may seem remarkably simple, but its effectiveness lies in understanding what makes for good statistical analysis: high quality data, and lots of it,” writes Shaw.

Classroom Activities: statistics, modeling

• (All levels) Shaw writes: “As far back as 1968, a statistical study was unable to find any link between shots, possession or passing moves and the outcomes of football matches.” Does this surprise you? Why or why not?
• (Mid level) In chess, the Elo rating (named after Arpad Elo) can be used to predict who will win a game. The Elo rating is even simpler than the model discussed in the article: It only depends on whether you win, lose, or draw games. Check out this table showing how the difference between your rating and your opponent’s rating corresponds to the chance that you’ll win a game.
• If your rating is 1800, and your opponent’s is 1600, what are the chances you’ll win?
• What if your rating is 1500 and your opponent’s is 1600?

—Leila Sloman

The Brain Uses Calculus to Control Fast Movements

Quanta Magazine, November 28

Much like a computer algorithm or instruction booklet, the fewer steps your brain takes to complete a task, the faster the results. And whether you’re a soccer player chasing a ball or a rat who just spotted a snack, you benefit from quick dialogue between your senses and muscles. In this article for Quanta Magazine, Kevin Hartnett discusses a new study of how the brain calculates motor control. Just thinking “stop” or “go” is not enough. “If you just take stop signals and feed them into [motor control region], the animal will stop, but the mathematics tell us that the stop won’t be fast enough,” a neuroscientist tells Hartnett. The real trick to quicker reflexes is about calculus.

Classroom Activities: calculus, rate of change

• (All levels) Introduce students to calculus with this video: How to use calculus in real life
• (Mid level) The article explains that movements depend on a “rate of change” between opposing inhibitory and excitatory signals. Imagine a simplified scenario where there is only one type of signal. In each pair below, choose which scenario has the largest rate of change.
• A signal of 100 units followed 1 second later by a signal of –99 units or A signal of 100 units followed 1 second later by a signal of 10 units
• A signal of 50 units followed 10 seconds later by a signal of 30 units or A signal of 50 units followed 1 second later by a signal of 30 units

—Max Levy

Is It Actually Impossible to “Square the Circle?”

Discover, December 16, 2022

A new paper has made progress on the “squaring the circle” problem, a question that dates back to the ancient Greeks. The modern form of the problem is to cut a circle into pieces and rearrange those pieces into a square, without any gaps or overlaps. (The ancient Greek version was to construct a square with the same area as a given circle using only a compass and straightedge. It is now known to be impossible.) This problem is difficult, but using advanced techniques, mathematicians have cut the circle into complex pieces that make it work. This year, András Máthé, Jonathan Noel, and Oleg Pikhurko found yet another way to square the circle, as Stephen Ornes writes for Discover. Still, there’s no easy solution: they use a mind-boggling number of pieces — around $10^{200}$ of them.

Classroom Activities: algebra, geometry, irrational numbers, pi, golden ratio

• (Algebra, Geometry) The ancient Greek problem of “squaring the circle” was impossible because $\pi$ is a kind of number called a transcendental number, as mathematician James Grime explains in a video for Numberphile.
• (Algebra, Geometry) Read this Quanta Magazine article by Dave Richeson about some other impossible problems in math.
• (Geometry) Show that if a cube has side length $1$, then a second cube with twice its volume has side length $\sqrt[3]{2}$. Based on the article, why does this make “doubling the cube” impossible?
• (Algebra) Richeson writes about a problem from ancient Greece involving the golden ratio $\phi = (1 + \sqrt{5})/2$. It can only be solved if there is a number $L$ for which $1/L$ and $\phi/L$ are both integers.
• Prove that $\phi$ is a root of the polynomial $x^2 - x - 1$.
• (Advanced) Is $\phi$ rational or irrational? What does this say about the ancient Greeks’ problem?

—Tamar Lichter Blanks

Mathematical Alarms Could Help Predict and Avoid Climate Tipping Points

Inside Climate News, December 27, 2022.

With effects of climate change already apparent, scientists fear the catastrophic effects of “tipping points” — points at which environmental damage creates a self-perpetuating cycle. In a recent paper, scientists studied how to predict when these tipping points will occur, reports Charlie Miller in this article. If this research can help stop humanity from crossing a tipping point, huge amounts of environmental destruction could be prevented. But as scientist Michael Oppenheimer told Inside Climate News, “​​Don’t expect clear answers anytime soon … The awesome complexity of the problem remains, and in fact we could already have passed a tipping point without knowing it.”

Classroom Activities: equilibria, climate modeling

• (All levels) Miller writes: “The study’s authors use the analogy of a chair to illustrate tipping points and early warning signals. A chair can be tilted so it balances on two legs, and in this state could fall to either side. Balanced at this tipping point, it will react dramatically to the smallest push.” Identify whether the following systems have tipping points, and if so, what they are:
• A ball rolling around on a hilltop
• A ball rolling around inside a bowl
• A rocket launching into space
• (Mid Level) For more on math and climate prediction, read “Climate modelling made easy”, by Chris Budd for Plus Magazine.

—Leila Sloman

How to win the gift-stealing game Bad Santa, according to a mathematician

The Conversation, December 15, 2022

If you’re a competitive person, you probably feel that the whole point of playing a game is to win — even at Christmas party games. In the game Bad Santa (also known as White Elephant or Yankee Swap), each person brings an anonymous gift, and gets a chance to open a gift from the pool of presents. The twist is that people can steal each other’s gifts. Some lucky players take advantage of this twist to snag their favorite item in the bunch, but others may end up with their least favorite. “It’s a good alternative to buying a gift for everyone, and a great way to ruin friendships,” writes Joel Gilmore, a mathematician from Griffith University who wrote about the game for The Conversation. If you want to win the best presents next year, it helps to understand favorable strategies. In this article, Gilmore describes running computer simulations of the game to find the most fair rules and the most successful strategies.

Classroom Activities: simulations, optimization

• (All levels) Play a simplified game of Bad Santa similar to Gilmore’s model. Form groups of 10 students. For the gift pool, take 10 cards from a regular card deck. Higher numbered cards represent better gifts. Choose a set of rules from the article and play the game. Students should feel free to use strategies discussed in the article as well.
• (Mid level) Discuss the results of the Bad Santa card game.
• Who feels their strategy worked, and why?
• Whose strategy did not work, and why?
• Who felt that they had no control over their win or loss and why?
• (Mid level) Change the rules and repeat. If the results are different, explain why you think the rules helped or hurt.

—Max Levy