Math Digests December 2023

One Futurama Plot Was So Complicated It Created A New Kind Of Math
Can Carbon Capture Live Up to the Hype?
Quantum computer sets record on path towards error-free calculations
The problem of thinking in straight lines
How Cryptographic ‘Secret Sharing’ Can Keep Information Safe

One Futurama Plot Was So Complicated It Created A New Kind Of Math

/Film, December 3, 2023.

An episode of the animated sci-fi sitcom Futurama features a machine that allows any two characters to swap bodies. The catch: once two people switch, the machine won’t work for that same pair of bodies again. They can’t switch back. At least, not directly. During the chaos of the episode, nine characters hop around to different bodies. But can they return things to normal? The answer is yes—but it requires some serious math. The show’s writer Ken Keeler, who has a PhD in mathematics, proved that it is possible to return everyone to their own bodies, as long as you add in two new participants who have not used the machine before. The story of Keeler’s result, also known as the “Futurama Theorem,” is described by Witney Seibold in an article for /Film. In the show, two members of the Globetrotters (a futuristic, math-genius version of the Harlem Globetrotters) discover the theorem, and return everyone to their original bodies to save the day.

Classroom Activities: group theory, symmetric group

(All levels) At the beginning of the episode, two people swap bodies. Call them Person 1 and Person 2. Suppose that Person 3 and Person 4 agree to help out by participating in some body swaps. Find a sequence of swaps so that no individual swap is repeated and everyone ends up in their original body. (Keeler’s theorem proves that this is possible.) Hint: This can be achieved with a total of 5 swaps. To keep track of the swaps, make a table whose columns represent the swaps as well as the four bodies (1, 2, 3, and 4), and where each row keeps track of the mind in each body, as below.
Fill in this table to figure out how to return Minds 1 & 2 back to the correct bodies.
- Suppose that only Person 3 agrees to help. Is there still a way to return everyone to their original bodies? Why or why not?
- (High level) The episode shows a sketch of Keeler’s mathematical proof, which is written using symmetric group notation. In this notation, (12) stands for the swap between Person 1 and Person 2, and a sequence of swaps is read right to left. For example, (23)(12) stands for the swap between 1 and 2 followed by the swap between 2 and 3. Try writing out your answer for four people using this notation. Then see if you can write down a solution for five people—that is, once three people have swapped around their bodies, find a way to fix things using two new participants.
(High level) As a class, work through the Futurama Theorem activity by Cheryl Grood featured on mathcircles.org.

—Tamar Lichter Blanks

Can Carbon Capture Live Up to the Hype?

The New York Times, December 6, 2023.

At a United Nations climate conference in December, policymakers debated the future of fossil fuels. Nonrenewable sources contribute 80 percent of the world’s energy supplies, and around 75 percent of greenhouse gas emissions. Fossil fuel emissions must go down to halt climate change, but oil and gas producers contend that technology can capture and sequester the harmful gases. But this scientific solution isn’t as simple as it sounds. “It would be nearly impossible for countries to keep burning fossil fuels at current rates and capture or offset every last bit of carbon dioxide that goes into the air,” Brad Plumer and Nadja Popovich write. “The technology is expensive, and in many cases there are better alternatives.” In this article, Plumer and Popovich analyze the limits of carbon capture technology, and what those limits mean for the future of energy production.

Classroom Activities: rates, data analysis, percentages

(All levels) Choose from these classroom activities about climate change and math from NRICH, including one that allows students to compare their carbon footprints, and another that asks students to design an efficient map for delivering energy by drawing a network that maximizes connections but minimizes overall length.
(Mid level) Approximately 15% of greenhouse gas emissions comes from livestock production, 30% come from automobile emissions, and 40% come from burning coal for energy. Which of the following would reduce total emissions the most?
- Reducing automobile emissions by 20%
- Reducing livestock production by 95%
- Reducing coal use by 40%
(Mid level) Suppose that the world emitted 50 billion metric tons of greenhouse gases (GHG) in 2023. Create a table based on the following to show how that number changes in the future.
- What will be the amount of annual emissions if the global emissions rate increases by 1.7% every year from 2023 through 2030?
- How many years would it take to get back to 50 billion metric tons per year if, starting in 2030, emissions reduce by a net 1% every year?
- Suppose that burning coal makes up 40% of total emissions and a new carbon capture technology promises to sequester GHG from coal emissions. However, in the first year, the technology can only sequester 0.5% of annual coal emissions. If in every subsequent year, the technology can capture 10% more than it did the previous year, how many years will it take for global emissions to decrease by 10 billion metric tons per year? (Begin your calculations from a total of 50 billion metric tons per year in 2023, and assume that all other emissions don’t change.)

—Max Levy

Quantum computer sets record on path towards error-free calculations

New Scientist, December 6, 2023.

In labs around the country, researchers are working on building computers unlike any you have ever used. Traditional computers store and process information as binary digits, or “bits,” 0 and 1. A quantum computer uses principles from quantum physics to carry out calculations at speeds far beyond traditional computers thanks to “qubits” which can be a combination of both 0 and 1 simultaneously. However, when researchers link together many qubits to create a more practical machine, computational errors become more common. In this New Scientist article, Karmela Padavic-Callaghan writes about a promising breakthrough that benefits from a special type of qubit. “Using thousands of rubidium atoms cooled to near absolute zero, they achieved a record-breaking 48 logical qubits simultaneously, over ten times the previous high,” Padavic-Callaghan writes. “This achievement marks a crucial step toward practical quantum computing.”

Classroom Activities: quantum mechanics, probability

(Mid level) Watch this PBS video about the mathematics of quantum computers.
- Explain in your words how a quantum computer works.
- What is special about the mathematics of a qubit compared to a traditional bit?
- In quantum computing, what role does probability play in the measurement of qubits, and how does it affect the outcome of computations?
- If you have a quantum system with 3 qubits, how many different possible states can it represent in superposition?
(Mid level) Read this article from IEEE Spectrum. If two qubits are entangled, the first qubit can be in a superposition of states $|0\rangle$ and $|1\rangle$, and the second qubit will always have the opposite state of the first one due to entanglement, how many possible states of these entangled qubits exist? What are the possible states?

—Max Levy

The problem of thinking in straight lines

BBC, December 31, 2023.

“Laura is a sprinter,” writes Kit Yates for BBC. “Her best time to run 100m (328ft) is 13 seconds, how long will it take her to run 1km (3,280ft)?” If you multiplied 13 seconds by 10 to answer this question, you’re not alone, says Yates. But view this as more than a multiplication problem with set dressing, and you might notice an issue with that strategy: While Laura’s 100-meter sprint time is well behind the times recorded in World Athletics list of top 100-meter times for women, a 130-second kilometer is unheard of. “The linear answer would see Laura utterly destroying the world record for running 1km,” writes Yates. That’s just one example of a scenario in which people may be inclined to assume, against the evidence, that things are linear. In this article, Yates argues that this “linearity bias” keeps us from viewing things accurately, with unwanted consequences.

Classroom Activities: nonlinear equations, mathematical modeling, exponentials, statistics

(All levels) On the Desmos graphing calculator, let the horizontal axis measure race distance and the vertical axis measure race time. Using data from World Athletics, plot the women’s world records for all races less than 1500 meters.
- Using these instructions, find a line of best fit of the form $y = ax + b$. How close can you get $R^2$ to 1? (Hint: Adjust the slider settings so that the step size is 0.001, and $a$ ranges between 0 and 0.5.)
- Now try finding a quadratic curve of best fit, of the form $y = ax^2 + bx + c$. Now how close can you get $R^2$ to 1? Assuming this equation applies to Laura’s running time, how fast would she run the 1000-meter race?
- According to François Labelle, a better formula has the form $y = ax^b$. Using this form, how close can you get $R^2$ to 1? According to this formula, how fast would Laura run the 1000-meter race?
- Do you think these formulas would apply to Laura? Why or why not?
- (High level) Using least squares, calculate the linear and quadratic equations of best fit, and compare the $R^2$ values.
(Mid level) Yates brings up the example of debt, which grows exponentially over time. Suppose you take out a $100 loan, and every year it accrues 20% interest.
- Without using a calculator or writing anything down, guess how much the debt will have grown after 10 years if you don’t pay anything back.
- How would you calculate how much you will owe after one year if you don’t pay back the loan? Find a general formula for how much you will owe after $n$ years.
- Using your formula, after 10 years, how much will you owe? Was your initial guess based on linearity bias?
- How long will it take for the debt to double?

—Leila Sloman

How Cryptographic ‘Secret Sharing’ Can Keep Information Safe

Scientific American, December 7, 2023.

In 1979, Adi Shamir came up with a technique which allows a group of people to crack a code, but only if they all work together. In this article, Manon Bischoff walks us through how it works, using a mother who wants to leave an inheritance to her five sons, but who also wants to make sure they split it fairly.

Classroom Activities: finite fields, fitting curves

(Mid level) Bischoff explains how in a simpler scenario with two or three sons, the woman could use linear or quadratic curves to make sure the sons have to work together to unlock the combination to her safe. We’ll repeat her example, with a quadratic curve instead of a linear one.
- Have students read the article to understand how this technique works.
- Now students will fit a quadratic curve of the form $y = ax^2 + b + c$. The combination to the safe is the coefficients $(a, b, c)$. Give them two coordinate pairs: $(1, 2)$ and $(-1, 0)$ are both on the curve. What can students figure out about the coefficients based on this information?
- After a few minutes, give the third coordinate: $(0, -2)$. This should be enough to calculate the combination.
- This technique works for any curve with three parameters, as long as you can solve for the parameters. Have students come up with their own combination and parametrized curve, and trade problems with a friend.
(Mid level) Bischoff notes that if you know the parameters of the curve are integers, that might make the technique less secure. Does this apply to the previous problem? Why or why not?
(High level) Shamir’s technique uses finite fields. Learn about the modular arithmetic of finite fields using this article from NRICH. Repeat the previous exercise mod 5. How does the arithmetic change?