Math Digests June 2023

When it comes to advanced math, ChatGPT is no star student

Ars Technica, May 20, 2023.

As students increasingly turn to AI for help with math homework, a worrisome risk emerges, potentially undermining their fundamental mathematical skills. In fact, ChatGPT wrote the first sentence of this paragraph, but the bot’s original sentence sounded awkward and needed editing. Although ChatGPT’s impressive-sounding answers have made students more comfortable offloading their homework, it’s not always easy to know when the chatbot makes a mistake. ChatGPT is trained on language data, which makes it more proficient with language. That means its pure math skills tend to be more vulnerable, writes Kenna Hughes-Castleberry in an article for Ars Technica. This article describes the disappointing advanced math skills of ChatGPT. “One should always double-check its outputs before trusting them blindly,” a mathematician told Hughes-Castleberry. “ChatGPT is not a magic tool that can solve any math problem, but it can be a helpful companion to give you some hints and suggestions.”

Classroom Activities: ChatGPT, AI tools

  • (All levels) Ask ChatGPT to list three examples of published academic articles about AI and math, along with the authors.
    • Discuss specific ways to verify whether ChatGPT gave you useful results.
    • Apply these ideas. Were the results useful? If not, why do you think ChatGPT failed, based on what you learned from the article?
  • (High level) The article mentions testing ChatGPT with six types of math problems using a dataset called GHOSTS. Using the academic article mentioned in the story, answer the following:
    • Describe in your words what each of these math problems evaluates.
    • Provide examples of each.
    • Prompt ChatGPT with an example for each and describe whether it does a good or bad job.
    • Based on these results, what are two ways in which you think ChatGPT could reliably help you with your math work?
    • What are two ways in which you definitely would not rely on ChatGPT for help? 

—Max Levy

First-Year Graduate Student Finds Paradoxical Set

Quanta Magazine, June 5, 2023.

Add any two numbers from the set ${ 1, 5, 11, 13 }$, and you’ll get a different sum. The sum $5+13$ gets you $18$, which is different from $1+11=12$, for example. A set of numbers with this property, where every pair of numbers in the set has a unique sum, is called a Sidon set. As another example, ${1, 2, 3, 4, 5}$ isn’t a Sidon set, since $1+5$ and $2+4$ both add up to $6$. In an article for Quanta Magazine, Alex Stone reports on a new preprint about Sidon sets by mathematics graduate student Cédric Pilatte. Pilatte proved the existence of an infinite Sidon set that has an additional special property: Every large number, past a certain point, is the sum of three or fewer numbers in the set. Until Pilatte’s paper, no one knew whether or not such a set existed. The problem had been left unsolved since it was posed in 1993 by Paul Erdős and two other mathematicians.

Classroom Activities: number theory, combinatorics, sets, polynomials

  • (All levels) Is $\{ 3, 5, 7, 9, 11 \}$ a Sidon set? What about $\{ 1, 2, 8, 20 \}$? Justify your answers.
  • (All levels) Come up with your own example of a Sidon set, as well as an example of a set that is not a Sidon set. Swap with another student and try to determine which of their two examples is a Sidon set.
    • (High level) If you want an extra challenge, find a Sidon set consisting of four numbers in which the largest number is $7$. Note: there are two different correct answers.
  • (All levels; Algebra) In the article, Stone writes that Pilatte replaced numbers with polynomials as part of the proof of his result. An irreducible polynomial is a polynomial that can’t be factored into simpler parts — that is, parts that have smaller exponents. The polynomial $x-5$ is irreducible, but $x^2-1$ is not, since $x^2-1 = (x-1)(x+1)$.
    • Discuss: The article says that irreducible polynomials are analogous to prime numbers. What do you think is the idea behind this comparison?
    • For each of the following polynomials, determine whether or not it is irreducible: $x^2 – 5x + 6$, $x^2 + 1$, $x^3 + 3x^2$. For each of the following integers, determine whether or not it is prime: $13$, $24$, $51$. How did you approach each of these two problems? (Note: For the purposes of this activity, all polynomials have coefficients in the real numbers.)
    • (Advanced) Is $x^4 + 1$ irreducible? Justify your answer.

—Tamar Lichter Blanks

Ancient plant’s leaves didn’t follow golden rule as modern ones do

New Scientist, June 15, 2023.

The golden ratio has been studied since the ancient Greeks. It’s well-known for its ubiquity in nature, such as in the arrangement of flower petals. In a recent study, researchers found that one plant from 400 million years ago displayed distinctly non-Fibonacci leaf arrangements. That’s not unheard-of in modern plants, including some relatives of the one studied in the new paper, reports Corryn Wetzel for New Scientist. But over 91% of today’s plants do conform to Fibonacci patterns. The new study suggests that Fibonacci spirals may be less intrinsic to plants than some researchers thought, says Wetzel.

Classroom Activities: Fibonacci numbers, algebra, golden ratio

  • (All levels) The golden ratio is closely tied to the Fibonacci numbers, a sequence where each term is the sum of the previous two terms. If $F_n$ is the $n$-th Fibonacci number, then $F_n = F_{n-1} + F_{n-2}$. The first ten Fibonacci numbers are 1, 1, 2, 3, 5, 8, and 13.
    • Compute the next 3 Fibonacci numbers.
    • The golden ratio is approximately $F_n/F_{n-1}$. Compute $F_n/F_{n-1}$ for n = 2, 3, 4, 5, and 6.
  • The golden ratio $\varphi$ was studied as far back as Euclid in his Elements, which described the foundations of geometry. Imagine you have a line split into two segments, one segment of length 1 and one of length $\varphi$. The ratio of $\varphi$ to the length of the full line is the same as the ratio of the two line segments to one another; that is, the ratio 1 to $\varphi$.
    • (Mid level, Algebra) Calculate $\varphi$. Does it match your calculations from the first exercise?
    • (High level, Geometry) A golden rectangle is a rectangle that can be split into two pieces: One piece is a square, and the other is another golden rectangle — that is, it’s smaller, but with the same proportions as the larger rectangle. Suppose the square part has side lengths 1. What are the side lengths of the golden rectangle?
  • (All levels) Read this article from IFLScience about the golden ratio.
    • (Mid level) In this Buzzfeed News article, writer Audrey Engvalson points out spirals and Fibonacci-like patterns in everything from nautilus shells to cabbages to a broken computer screen. Think of some strategies for how to tell which examples are truly showing the golden ratio versus some other pattern.
    • Read the Buzzfeed article. Using the strategies you came up with, try to identify the Fibonacci numbers or the golden spiral in each example. How many can you identify? Which examples do you think are exhibiting a different pattern?

—Leila Sloman

Gift Wrapping Five Oranges Has Outwitted the Best Minds in Mathematics for Generations

Scientific American, June 6, 2023.

Math reserves its strangest behavior for the dimensions that we can’t perceive. Take sphere-wrapping: In the late 1800s, Norwegian geometer Axel Thue studied how to arrange circles such that they could be wrapped by the shortest possible length of imaginary string. Since mathematicians have long studied how problems change when taken to new dimensions, extending this problem to gift-wrapping spheres was a no-brainer. “It will probably come as no surprise that the three-dimensional case raises even more questions than optimal circular packing in the two-dimensional world,” writes Manon Bischoff. But as Bischoff describes in this article, the highest dimensions, far past 3D, are weirder yet, bringing about surprising “catastrophes” in mathematical understanding.

Classroom Activities: area, volume, higher dimensions

  • (Mid level) Suppose you have four round coins with radius r. Write expressions for the minimum length of string needed to enclose the following:
    • Four coins arranged in a row
    • Four coins arranged in a square (two evenly stacked rows)
    • Four coins arranged in a diamond. Compare and explain the results. Which length of string is longest? Which is shortest?
  • (Mid level) Consider a related problem that spans multiple dimensions. Draw four circles of radius of 1 inside a square with sides of length 4.
    • What is the radius of the largest circle that can fit in the gap at the center?
    • Now, in three dimensions, sketch a cube of side length 4 filled with eight spheres of radius 1. What is the radius of the largest sphere that can fit in the center gap?
  • (High level) Notice the pattern inside the square root term that appears in the radius expression for 2D and 3D. This trend continues for each dimension (e.g. 4D, 5D, 6D … ).

—Max Levy

Has a mathematician solved the ‘invariant subspace problem’? And what does that even mean?

The Conversation, June 11, 2023.

Over 35 years ago, Per Enflo answered a famous open problem known as the invariant subspace problem. But his proof left mathematicians wondering about a special edge case — a case which Enflo recently announced he’d taken care of, in a preprint he posted online in May. In this article for The Conversation, Nathan Brownlowe, a lecturer at the University of Sydney, explains the invariant subspace problem and Enflo’s history with it.

Classroom Activities: linear algebra

  • (High level, Algebra 2) The official version of the invariant subspace problem concerns a highly abstract type of mathematical space called a Banach space. But there’s a much more concrete version of the problem that involves matrices and vectors.
    • Refresh your matrix multiplication with these practice problems from Khan Academy.
    • Show that in the following examples, $MV = 2V$.
      • $M = \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}, V = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$
      • $M = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}, V = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$
      • $M = \begin{bmatrix} 0 & 0 \\ 1 & 2 \end{bmatrix}, V = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$
    • (High level) Let $M = \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}$. Draw the line through the origin that includes the point $V = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$. If you applied $M$ to the whole line, how would it transform?
      • What if you applied $M$ to the line y = x/2? How would it transform?
      • The line that includes $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ is called an invariant subspace of M. In what sense is the line invariant? Can you find another line that makes an invariant subspace?

—Leila Sloman

Some more of this month’s math headlines: