Math Digests April 2023

A Number System Invented by Inuit Schoolchildren Will Make Its Silicon Valley Debut

Scientific American, April 10, 2023.

The newest version of Unicode will allow you to type using an Inuit number system called “Kaktovik”. Kaktovik was developed in the 1990s as part of a class project, and draws on the counting practices of Iñupiaq, the language of the Alaskan Inuit. Kaktovik is “strikingly visual”, writes Amory Tillinghast-Raby for Scientific American. Performance on standardized math tests soared after it was implemented in middle schools in 1997. But in many schools, Kaktovik became a casualty of the No Child Left Behind Act. Its inclusion in Unicode may help bring it back to life.

Classroom Activities: arithmetic, counting

  • (All levels) Study the “Kaktovik numerals” chart from the article in class. Write out the following numbers using Kaktovik: 5, 11, 25, 33, 99.
  • (Mid level) Now study the chart “Solving equations with the Kaktovik System.” Using only Kaktovik numerals, calculate the following:
    • $25 – 5$
    • $99 \div 33$
    • $25 \div 5$
    • $99 + 11$
  • (Mid level) Now try the same calculations, this time using the Hindu-Arabic number system. Reflect: Which calculations did they find easier? What did they notice while doing the calculations?
  • (High level) Another type of number system involves using Hindu-Arabic numbers, but with a different base. The binary system is a base-2 number system. Read this online lesson and do the questions under “Your Turn”.
    • Kaktovik uses a base 20 system. Convert the Hindu-Arabic numbers from the previous exercises to base 20, and repeat the calculations in base 20.

—Leila Sloman

Hobbyist Finds Math’s Elusive ‘Einstein’ Tile

Quanta Magazine, April 4, 2023.

Sometimes being a good mathematician is all about having a knack for finding patterns, and sometimes it’s about showing there are no patterns at all. That was the case in a recent breakthrough in tiling, the study of how to arrange shapes so that they completely cover a surface without any overlaps or gaps. Tiling usually creates repeating patterns, like the honeycomb pattern that hexagons make. But for decades, mathematicians and hobbyists searched for a more elusive phenomenon: one shape that can tile a plane without creating a pattern. (A German geometer dubbed the hypothetical tile an “einstein” — a pun on the German phrase “ein stein,” which means “one piece.”) Recently, puzzle enthusiast David Smith found an einstein. “We were all pretty blown away,” one computer scientist told Erica Klarreich, who tells the story of how Smith finally cracked the code with a shape that looks like a hat.

Classroom Activities: tiling, periodicity

  • (All levels): Print and cut out 20 identical equilateral triangles, 20 identical pentagons, and 20 identical hexagons. Then answer the questions below.
    • Using your cut-outs, try to tile a plane with each of the three shapes.
    • Which shapes tile the plane?
    • If a shape does not tile the plane, do you think that changing the length of its sides will help? Why or why not?
  • (Mid level): Watch this Veritasium video about Penrose tiles. Print out 20 copies of kites and 20 copies of darts. Are you able to arrange the tiles in a repetitive pattern? Based on the video and your experimenting, how many different patterns and arrangements do you think are possible? (Infinity is a possible answer!)
  • (High Level) Read Klarreich’s article about the hat tile. Klarreich writes that “the hat is one of infinitely many different tiles of this type.” Based on what you have read, why are there an infinite number of “hat tiles” possible?

—Max Levy

How Much Detail of the Moon Can Your Smartphone Really Capture?

Wired, April 7, 2023.

In this article for Wired, Rhett Allain answers a simple question: Why can’t you point a camera at the moon, zoom in, and walk away with a photo that captures the moon’s intricate topography? The answer has to do with how light behaves as it squeezes through the small opening of a camera lens. Light waves far away from the lens are “diffracted,” or spread out. As a result, light waves coming from different points on the moon overlap with one another, making them impossible to distinguish. The upshot? You’ll never find a smartphone camera good enough to get a clear, detailed photo of the moon — at least, not unless you make use of some fancy image correction software. “It’s not a limit on the build quality of the optical device; it’s a limit imposed by physics,” says Allain.

Classroom Activities: optics, trigonometry

  • (Mid level) Allain writes that whether you can distinguish between two points on the moon depends on their angular separation. That is, draw a line from your eye to Point A, and another from your eye to Point B. The angle between these lines determines whether Points A and B are distinguishable. Try the following exercises.
    • Imagine you are looking directly at two objects, A and B, which are 10 meters away from you. A and B are about 10 centimeters apart from each other. According to the formula found in the article, what is the angular separation between A and B?
    • What if A and B are one meter apart? Five meters?
  • (High level) In his formula $\theta = h/r$, Allain is taking advantage of a common approximation: $\sin(\theta) \approx \theta$. Calculate the angular separations from the previous exercise precisely using the inverse sine function. How do the answers compare?
  • (High level) Allain’s smartphone camera has a lens about 0.5 cm in diameter. Suppose Allain wanted to take a photo of two spots of light that were 1 meter apart, and the wavelength of the light was 500 nm. How far away could those objects be before they would be impossible to distinguish?

—Leila Sloman

Mathematician Breaks Down the Best Ways to Win the Lottery

Wired, April 3, 2023.

If you like to buy lottery tickets, mathematical thinking can help you increase your potential winnings. In a video from Wired, mathematician Skip Garibaldi explains lottery strategies that can theoretically help you win more money. You can try choosing unpopular numbers to increase the odds that you’ll be the only person who wins the jackpot, or carefully select state scratch-off games with a lot of unclaimed prizes. There have even been rare cases of lotteries where people were able to consistently profit, but as Garibaldi explains, it’s usually a losing bet: “It’s really hard and unusual to be in a situation where you could reasonably expect to make money on the lottery.”

Classroom Activities: probability, combinatorics

  • (Probability; All levels) Imagine a small lottery with just one prize, worth \$70. There are 100 tickets and each ticket costs \$1. At the drawing, one of the 100 tickets will be selected randomly to win the prize.
    • If you buy one ticket, what is the probability that you will win the prize?
    • If you buy all 100 tickets, what is the probability that you will win the prize? In this situation, how much money would you spend, and how much would you receive?
    • The expected value of a lottery ticket is the amount you would expect to win, on average, if you played the same lottery many times in a row. (See this summary of the concept of expected value.) To compute the expected value, take the amount of money you would receive if your ticket won the prize, and multiply that by the probability that that ticket will win. What is the expected value of a ticket in this lottery?
  • (Probability; High level) In the video, Garibaldi talks about a type of lottery that you win by correctly guessing a 4-digit number. He discusses a particular kind of entry in this lottery called a “six-way box.”
    • The winning “ticket” in this lottery is a 4-digit number. How many different 4-digit numbers are there, including numbers that start with 0?
    • A six-way box is a type of entry in this kind of lottery, where you choose two numbers—say, 1 and 2—and bet on all of the 4-digit numbers that have two of each of those digits, such as 1122 and 2121. How many combinations of two 1’s and two 2’s are there? Justify your answer.
    • If you bet a six-way box, what is the probability that you will win this kind of lottery? (Garibaldi answers this question in the video: start watching at time 5:26.)

—Tamar Lichter Blanks

The Wondrous Connections Between Mathematics and Literature

The New York Times, April 7, 2023.

Mathematics is the language of the universe. It lets us communicate about intangible forces in physics, and double the amount of flour in a cake recipe. And it also shows up in literature, as mathematician Sarah Hart notes in an article for the New York Times. “Leo Tolstoy writes about calculus, James Joyce about geometry. Fractal structure underlies Michael Crichton’s ‘Jurassic Park’ and algebraic principles govern various forms of poetry,” Hart writes. “We mathematicians even appear in work by authors as disparate as Arthur Conan Doyle and Chimamanda Ngozi Adichie.” In this essay, Hart admires the symbiosis between math and writing.

Classroom Activities: cycloids, the universal language

  • (All levels) Based on the essay, sketch what a cycloid should look like for a wheel that is 2 inches in diameter and traveling at 1 inch per second. Would this shape change if the speed was 4 inches per second? If so, how?
  • (Mid level) Have each student bring in their favorite book. Then form small groups, and discuss the following questions.
    • Can you remember (or find) a reference to mathematics in your book? Why does the author include this reference (e.g. what do they find interesting enough to write about)?
    • Math also influences literature in a more subtle way than explicit references. Choose a paragraph or two from the book without any quotes, and read it aloud to the group. Then count the number of words in each sentence. Are they all about the same? How do they vary? Do you notice anything else? Compare the results for each group member’s book.
    • Based on what you read, discuss why the number of words in sentences might matter. (Hint: Think about style, emotion, and readability.)
  • (High level) Hart writes, “It’s worth pointing out as well that the links between mathematics and literature do not run in just one direction.” In what ways is mathematics similar to a language, like English?

—Max Levy

Some more of this month’s math headlines: