Math Digests October 2024

Why Democracy Lives and Dies by Math

The New York Times, October 24, 2024.

Democracy is “very much mathematical,” Ismar Volic told the New York Times in this Q&A. Volic, a mathematician at Wellesley College, is featured in a new documentary about the mathematics behind civil rights, democracy, and policy. For this article, writer Siobhan Roberts spoke to Volic and the documentary’s director, Vicki Abeles. “Decisions are increasingly driven by data, by algorithms, by statistics,” Abeles told Roberts. “You need a certain amount of math to be able to fully participate as a citizen.”

Classroom Activities: voting, gerrymandering, policy

  • (All levels) Read the article, then answer the following questions.
    • Why do Abeles and Volic believe math is necessary to participate in today’s democracy? Do you agree? Why or why not?
    • Brainstorm a list of issues that citizens need math to fully understand. What type of math is necessary?
    • Brainstorm a list of issues that government leaders need math to fully understand. What type of math is necessary?
  • (Mid level) Volic told the New York Times that in his class “Math and Politics,” students “are upset that there is a path to win the presidency with only 23 percent of the popular vote, that gerrymandering is rampant, that the system silences and disenfranchises millions of people.”
    • Read about the debate on the Electoral College. Write out your initial thoughts on this debate.
    • Read the linked NPR article on how few votes a candidate can receive and still win the Electoral College.
    • The data NPR used is now 12 years old. Look up the most recent available data on voting turnout and distribution of electors and repeat their calculations to find out the smallest percent of the popular vote a winning candidate could have received in 2024. (For a more detailed description, NPR references this video.)
    • Write out your thoughts on the Electoral College after doing this exercise. Did your opinion change? What else do you want to learn?

—Leila Sloman

Deep inside a Norwegian fjord, a dream of farming salmon sustainably

NPR, October 13, 2024.

One-fifth of the United States’ salmon comes from Norway. The Scandinavian country’s water has the right salinity and temperature to attract the fish, and strong currents keep the water fresh. However, critics say that Norway’s salmon farms damage the environment of the narrow coastal inlets, called fjords, that house farming operations. Farmed salmon escape and take over wild spawning zones; they also facilitate a booming population of parasites known as sea lice. This NPR story describes a sustainable solution to Norway’s salmon population problem.

Classroom Activities: population modeling, algebra

  • (All levels) Based on the article, what are four environmental problems created by salmon farming in Norway?
  • (Mid level) For the following questions, assume that at the start of the first year ($t=0$) there are 200,000 farmed salmon ($F$) and 500,000 wild salmon ($W$).
    • What percent of the total salmon population in open water is farmed?
    • Norway farms an estimated 850 million salmon per year. If 200,000 escape per year, what percent of the farmed population escapes?
    • To sustain wild populations, wild salmon must survive in open water and breed successfully. Farmed salmon are less adapted to survival in the wild. If wild salmon populations can grow by a factor of 1.5 per year (50% population growth per year) and farmed salmon populations by a factor of 0.5 (50% population loss per year), write algebraic expressions to model the number of wild salmon over time, $W(t)$, and the number of farmed salmon over time, $F(t)$.
    • Assuming no crossbreeding, what is the population of each after five years?
    • If 30% of farmed salmon mate with wild salmon, and the hybrids have a reproduction rate of 0.9, write and discuss how you would write algebraic expressions to model the population. Is it possible to predict salmon populations based on these expressions by hand? Why or why not?

—Max Levy

Mbappe revealed as football’s next GOAT by Oxford mathematician

Irvine Times, October 23, 2024.

In 2021, mathematician Tom Crawford introduced an algorithm for scoring the overall greatness of soccer players. He called it the G.O.A.T. Index (standing for “Greatest of All Time”), and his findings decreed that Cristiano Ronaldo was the greatest player of all time. Now, he has come up with G.O.A.T. 2.0, for finding who the future G.O.A.T. will be among 10 early-career players. It predicts that Ronaldo’s successor will be Kylian Mbappé of Real Madrid.

Classroom Activities: data analysis, mathematical modeling

  • (Mid level) Watch this video where Crawford explains the index. Write, in your own words, a sketch of how it works.
  • (Mid level) Come up with your own G.O.A.T. Index. Your index can be about whatever you like — greatest actor of all time, greatest apple pie of all time, greatest video of all time, just to name a few possibilities — but it should assign an overall score to contenders based on a few relevant categories or attributes.
    • In detail, describe your index. What categories or attributes will you consider? How will you assign points within each category? How will you objectively choose contenders for the G.O.A.T.? Make sure to figure out as many details as you can before finding out any results! (Why is that important?)
      • As much as possible, explain the reasoning behind your choices.
    • Identify your contenders and apply your model. List the scores and rankings in each category as well as the overall scores and rankings.
    • Reflect: Did the results surprise you? What could be changed about your method, and would that change the results? Would you change anything if you were to do it again?
  • (All levels) Is Crawford’s G.O.A.T. Index the final word on who is the best soccer player of all time? Why or why not?

—Leila Sloman

Why This Great Mathematician Wanted a Heptadecagon on His Tombstone

Scientific American, September 12, 2024.

Johann Carl Friedrich Gauss was 18 when he solved a geometry problem that had eluded mathematicians for 2,000 years: How to draw a perfectly regular 17-sided polygon (a “heptadecagon”) with only a compass and straightedge. Constructing any regular polygon with just these simple tools requires some clever work. It helps to know that interior angles always sum to a circle’s 360 degrees (so an equilateral triangle has three interior angles of exactly 360/3 = 120 degrees). But an angle of 360/17 degrees is less intuitive to construct. This Scientific American article walks readers through regular polygon constructions and explains the significance of the heptadecagon.

Classroom Activities: geometry, straightedge, compass

  • (All levels) Follow along by doing each example shown in the article with a straightedge and compass:
    • Overlapping circles,
    • Equilateral triangle,
    • (Mid level) Square and hexagon.
    • (Mid level) Explain in your own words why the heptadecagon could not be constructed from the rules used above.
  • (Mid level) Calculate the number from Gauss’s equation to four significant figures. Use this number to construct a heptadecagon.

—Max Levy

Big Advance on Simple-Sounding Math Problem Was a Century in the Making

Quanta Magazine, October 14, 2024.

Last November, a mathematician in Chile proved a result that baffled researchers, including himself. Consider the sequence of numbers $n^2 + 1$, where $n$ is a whole number. The sequence begins 2, 5, 10, 17, 26 and continues infinitely. “The $n^2 + 1$ sequence offers a good starting point for investigating the relationship between addition and multiplication,” wrote Erica Klarreich for Quanta Magazine. As numbers in the sequence grow, the prime factors seem to grow as well. This new proof shows that the prime factors grow more quickly than ever reported before. This article tells the story of the new proof and describes its implications.

Classroom Activities: sequences, prime factors

  • (Mid level) The first 5 numbers in the sequence are 2, 5, 10, 17, 26. Write the next 5 digits in sequence and answer the following questions.
    • What arithmetic pattern do you notice between them? (Hint: How much larger is each number in the sequence compared to its predecessor?)
    • Based on this other rule, write a recursive rule that creates the sequence $n^2 + 1$.
    • Does a version of this rule also apply to a sequence of $n^2$?
  • (Mid level) Describe in your words what a prime factor is.
    • Write down the prime factors of the following numbers: 16, 37, 65
    • Does every prime factor for the $n^2 + 1$ sequence increase at every step?
    • What does the “growth rate” described in the article refer to? Discuss why Pasten’s growth rate is an improvement over previous results.

—Max Levy

Some more of this month’s math headlines