Math Digests November 2021

What Hot Dogs Can Teach Us About Number Theory

Quanta Magazine, November 18, 2021

No hot dog is complete without a bun. So it’s maddening that you typically have to buy eight buns at a time, but you can’t buy exactly eight hot dogs. Getting one package of each will leave you with two extra hot dogs and no buns—useless leftovers. Patrick Honner explains how the Chinese Remainder Theorem helps you avoid this situation. If you buy the right number of packages, you’ll end up with exactly one bun for each hot dog. Even when there are a few extra hot dogs or buns hanging out in the fridge, the Chinese remainder theorem guarantees you’ll be able to make things work out. If that’s too much math for you, there’s also an easier solution: In my experience, veggie sausages come in packages of four.

Classroom activities: number theory, modular arithmetic, cryptography

(High school) Have students read the article and complete the exercises at the end.
(High school) Honner mentions how the Chinese remainder theorem comes into cryptography. Have students use the A1Z26 cipher to decrypt codes in class:
- Break students into pairs. Have each student think of a secret word and encode it using the A1Z26 cipher. Then, have them encode it even further by finding the numbers mod 4 and mod 9. Thus the word CODE, encoded as (3-15-4-5), becomes (3-3-0-1) mod 4 and (3-6-4-5) mod 9.
- Have students trade their codes mod 4 with their partner and try to break them. After 5 minutes, allow them to share the code mod 9. According to the Chinese remainder theorem, they should now have enough information to break the code.
- Repeat the activity with new secret words, but this time have students share the code mod 9 first. Have students write about which way was easier and explain the reasoning they used.

—Leila Sloman

Redistricting in N.C.: New maps approved, favoring GOP

Spectrum News 1, November 4, 2021

Every 10 years, all the states in the US go through the process of redistricting: drawing new lines for congressional and state legislative districts based on the latest census data. The current redistricting cycle has been extremely contentious, with partisan map-drawers in many states creating maps heavily biased in favor of their political party. North Carolina is a perfect example. When mathematician Jonathan Mattingly’s team compared the state’s proposed maps to tens of thousands of alternatives, “We found that the map that has been proposed for the North Carolina House really dramatically under-elects Democrats. We have similar analysis for the Senate.” The mathematicians’ methods, which have already featured prominently in court cases in recent years, provide rigorous quantitative evidence that the maps in North Carolina and elsewhere intentionally give one party an unfair edge. (A Raleigh News & Observer article also quotes Mattingly.)

Classroom activities: gerrymandering, voting, geometry, Markov chain Monte Carlo

(Middle school) Teach students about gerrymandering with this lesson plan from Illustrative Mathematics. The lesson begins with simple examples of elections for a school mascot and school board, then gives students the opportunity to draw district maps that favor one candidate over another.
(Middle school / high school) Play this gerrymandering game by the Julia Robinson Mathematics Festival with 28 levels that increase in difficulty. Have students work in groups, then share their solutions with the class.
(High school) Teach students about gerrymandering with this lesson plan from Corwin. The worksheets include problems related to contiguity, the efficiency gap, and two mathematical measures of compactness.
- (Advanced) Introduce the concept of Markov chain Monte Carlo methods, which underlie the mathematical analysis of the North Carolina maps mentioned in the article.

Related Mathematical Moments poster and interview: Countermanding Gerrymandering.

—Scott Hershberger

How an oyster builds a perfectly round pearl

Science News, November 10, 2021

At the center of an oyster is something beautiful: a pearl. At the center of a pearl is something less so: a misshapen lump of debris. For years, scientists have wanted to figure out how oysters grow perfectly round pearls over irregular bits of junk. Now, the mystery is solved, Rachel Crowell writes. The answer is a mathematical pattern found commonly in nature, called 1/f noise or pink noise. Oysters build tiny layers of mineral and protein, called nacre, around the debris or grains of sand, gradually smoothing out the odd shape. Each layer of nacre has a different thickness that depends mathematically on the one below it. This inversely proportional relationship also shows up in seismic activity, classical music, and heartbeats.

Classroom activities: pink noise, inverse proportionality, waves

(All levels) Hear what pink noise sounds like on Wikipedia, and watch videos about pink noise in human biology and music production.
(All levels) Discuss why this sound is called pink noise. (Hint: sound and light are both waves. What’s the difference between white light and pink light?)
(Middle level) When teaching students to graph equations like $y=1/x$, use pink noise as an example application. (Note that most graphs of pink noise appear linear because the axes are on a logarithmic scale.)
(High level) What color might you associate with a sound that has a high intensity at middle frequencies but zero for high and low frequencies? What about a sound that has a low intensity at low frequencies but a high intensity at high frequencies? (Answers in the “Colors of Noise” Wikipedia page)

—Max Levy

Pythagoras’ revenge: humans didn’t invent mathematics, it’s what the world is made of

The Conversation, November 21, 2021

Is math real, or is it just a way of describing the world? According to Sam Baron, an Australian professor, math is as real as the paper, the pen, and the brain with which you do it. Baron recently wrote an article for The Conversation in which he argues that math is an essential part of nature, not just a way of describing it. He draws upon examples of math in nature to prove his point. Bees make six-sided cells to store honey since hexagons form the most efficient tiles. Periodic cicadas evolved to emerge every 13 and 17 years because prime numbers help them avoid the more regular feasting patterns of predators. “The world has two parts, mathematics and matter,” Baron writes. “Mathematics gives matter its form, and matter gives mathematics its substance.”

Classroom activities: tiling, prime numbers, nature

(All levels) Hexagons can tile a flat surface (cover it entirely with no gaps) to infinity. According to the honeycomb conjecture in math, the hexagon is the best shape for tiling a plane, in the sense that hexagons require the shortest total perimeter to cover a given area. Bees use this fact to build efficient honey storage.
- Watch this It’s Okay to Be Smart video called “Why Nature Loves Hexagons.”
- What other regular polygons can tile the plane? Using sturdy paper like construction paper, cut out an equilateral triangle, a square, a pentagon, a hexagon, and an octagon. Use each shape to sketch a tiling of a plane (if you can!).
- (Upper level) Once you’ve finished your tilings, measure the total perimeter and area of each design. Can you see why bees benefit from using hexagons?
(All levels) Suppose that a group of cicadas emerges every 13 years, and their two predators have lifecycles of 3 and 4 years.
- How often will the cicadas face the threat of the 3-year predator?
- What about the 4-year predator?
- How often will the cicadas face both predators in the same year?
- How would the answers change if the cicadas emerged every 10 years instead?
(High school) Using Baron’s article and this TED-Ed video to start the conversation, discuss whether you think math is invented or discovered.

—Max Levy

A 14-year-old won a prestigious award for his discoveries on ‘antiprime’ numbers

NPR, November 2, 2021

Since 2011, middle school students have entered their science projects in the Broadcom MASTERS competition in hopes of a $25,000 prize. This year for the first time, a young mathematician won the contest. Nell Clark covered Akilan Sankaran’s accomplishment for NPR. Akilan, who is 14, came up with an algorithm to quickly find “anti-prime” numbers, numbers with many prime factors. To do this, he used a function whose output depended on an integer’s factorization properties. In the end, the program he wrote delivered mind-bogglingly large anti-prime numbers in a short time. This isn’t just a mathematical curiosity: “highly divisible numbers are useful in computing because they can be used to divide data among computer processors, Akilan explains.”

Classroom activities: number theory, prime factorization

(Middle school) The mathematical definition of an anti-prime (or highly composite) number is one that has more divisors than any smaller whole number has. Ask students to find all seven anti-prime numbers from 1 to 40 based on this definition (without looking at the preceding link!). (Hint: 1, 2, and 4 are the first three)
(Middle school) Assign this prime factorization practice from Khan Academy.
(High school) For his project, Akilan came up with two functions, $f_s(n)$ and $S_k(n)$ (slide 3), that he used to find anti-prime numbers. He also compared them to the divisor function $d(n)$. On slide 5, he shares some properties of these functions.
- Have students prove the divisor function properties in the first and second rows of the table on slide 5: $d(p) = 2$ for any prime $p$, and $d(mn) = d(m)d(n)$ when $m$ and $n$ don’t share any prime factors.
- Have students give an example to show that if $m$ and $n$ share a prime factor, then the multiplicative property of $d$ fails (that is, $d(mn)\neq d(m)d(n)$).
- (Very hard) Have students try to prove the smooth function properties in the first and second rows: $f_s(p) = 1+ 1/p$ when $p$ is a prime, and $f_s(mn) = f_s(m)f_s(n)$ when $m$ and $n$ don’t share any prime factors.