Math Digests January 2023

A New Puzzle Turns Earth Into a Rubik’s Cube, but More Complex

The New York Times,  January 1, 2023.

The British-American mathematician and artist Henry Segerman can’t visualize shapes or scenes with his eyes closed. Yet this “aphantasia” seems to motivate him to bring math out of imagination and into his hands: as puzzles and tools. Last year, Segerman debuted a new puzzle called Continental Drift. It’s a sort of spherical blend between a Rubik’s Cube and a sliding puzzle. The puzzle represents Earth, and its surface is split into 12 pentagons and 20 hexagons like a soccer ball. To beat the game, players must slide the hexagonal tiles until they have recreated the correct map of the globe. It’s a complex game — Segerman has calculated that there are $7 \times 10^{31}$ possible arrangements of the tiles. (A Rubik’s Cube has one-trillionth as many.) This article by Siobhan Roberts describes Continental Drift as well as Segerman’s other mathematical creations from 2022. 

Classroom Activities: holonomy, math puzzles

  • (All levels) Watch the videos of Segerman’s various puzzles and inventions. Which is the most interesting to you and why?
  • (Mid level) As Roberts writes, the mathematical trick that complicates this game is called holonomy. If you slide the unique hexagonal tiles (and the vacant space) to loop around a pentagon between them, each tile will arrive slightly rotated. You must keep looping 5 more times to recreate the starting arrangement. 
    • Suppose you are moving 7 unique octagonal tiles and an empty space clockwise around the perimeter of an octagonal tile. How many moves would it take to return to the start configuration? (Hint: watch Segerman’s video that explains this holonomy math.)
  • (High level) Segerman studies topology, the study of geometry that ignores lengths or angles. Watch this video explaining a classic topology joke: “A topologist is somebody who can’t tell the difference between a coffee mug and a donut.” That’s because a coffee mug has one hole (the hole created by its handle) and so does a donut.
    • If a donut is “the same” as a coffee mug, which of the following pairs of shapes would also be considered the same:
      • Needle and Macaroni
      • Pretzel and College-ruled Paper
      • Gold brick and Bowtie tied in a knot

—Max Levy

Jumping beans’ random strategy always leads to shade — eventually

Science News, January 5, 2022.

Though jumping beans are really nothing but seed pods, they house moth larvae who prefer a cool, shady spot. When it gets hot, the larva inside starts to move, and the jumping bean jumps along with it. A new study analyzes the motion of jumping beans, concluding that — unable to see where it’s going — the jumping bean follows what’s called a random walk. As James Riordon reports for Science News, this means that if it keeps jumping around indefinitely, the jumping bean is sure to find a spot of shade.

Classroom Activities: probability, random walk

  • (Mid level) The jumping bean “walks” around on a two-dimensional surface, but you can also define a one-dimensional random walk: At each step, you choose randomly to go either one foot left or one foot right. Each option has probability ½.
    • What is the probability of taking 2 steps left, then one step right?
    • What is the probability of taking 2 steps left and one step right, in any order?
    • What is the probability that after 10 steps, you’re less than 9 feet away from the starting point?
    • How do your answers to the first three questions change if the probability of going left is ⅓, and the probability of going right is ⅔?
  • (High level) As Riordon explains for the jumping bean, if you follow the one-dimensional random walk forever, you’ll explore the whole number line. For instance, you’ll eventually be 1000 feet to the right of the starting point. Does this surprise you? Why or why not?
  • (All levels) For a more in-depth lesson on random walks, try Ralph Pantozzi’s award-winning lesson plan. In this 2-day lesson, students spend the first day collecting data on the random walk by participating in a “flip trip”. On the second day, students will examine the probability model in more detail.

—Leila Sloman

‘Warm, kind, intellectually brilliant’: Mathematician Martin Davis dies at 94

The Daily Californian, January 8, 2023.

Martin Davis, who made important contributions to mathematics and computer science, died on January 1, 2023. Davis became one of the earliest computer programmers after receiving a mathematics Ph.D. from Princeton University in 1950. As Ani Tutunjyan writes in an obituary in The Daily Californian, Davis is famous for his work on Hilbert’s tenth problem, one of 23 unsolved problems mathematician David Hilbert posed in 1900. Hilbert’s tenth problem is to find an algorithm that can take any polynomial equation with integer coefficients, such as $x^2 + 3y^3 – 4 = 0$, and determine whether or not that polynomial has any integer solutions. The algorithm should be able to give a yes or no answer for any polynomial after a finite number of steps. Thanks to the work of Davis and other mathematicians, it is now known that Hilbert’s tenth problem is unsolvable: it is impossible to find an algorithm that checks for the existence of integer solutions.

Classroom Activities: integers, algebra, polynomials, algorithms

  • (Algebra) The integers are the positive and negative whole numbers, along with zero. They include $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$, but not fractions like $\frac{1}{2}$ or irrational numbers like $\pi$. 
    • Find an integer solution for $x^2 – 4 = 0$. In other words, find an integer $a$ with the property that $a^2 – 4 = 0$.
    • Find integers $a$ and $b$ with the property that $a + 4b – 17 = 0$. Discuss your answer with another student. Did you find the same solution, or two different ones? Can you find another solution?
    • (High level) Explain why $2x – 1 = 0$ does not have an integer solution.
    • (High level) Explain why $x^4 + y^2 + 5 = 0$ does not have an integer solution. (Hint: If $a$ and $b$ are integers, can $a^4 + b^2$ be a negative number?)
  • (All levels) An algorithm is a list of steps that starts with some kind of input and produces a result after a finite number of steps. The quadratic formula is a kind of algorithm: it is a process for taking any quadratic equation $ax^2 + bx + c = 0$ and finding its solutions.
    • Read the examples of polynomials with integer coefficients listed on the Wikipedia page for Diophantine equations. Do you find it surprising that there is no algorithm that can always check whether or not a polynomial has integer solutions, as Davis and other mathematicians have shown? Why or why not?
    • Watch the introduction to algorithms video on Khan Academy, then try the guessing game. Try to answer the question after the longer guessing game: Why should you never need more than 9 guesses?

—Tamar Lichter Blanks

Can you really drive around India in an EV? Notes from an IIT engineer’s 8,000-km journey

Conde Nast Traveler, January 3, 2023.

Electric vehicles are popular among drivers who want to reduce their carbon footprint. But the thought of embarking on a long journey can be intimidating due to concerns about battery range. In an article for Conde Nast Traveler, an EV owner describes journeying through India in an electric car to test the feasibility of using an EV for long-distance travel. The trip covered approximately 8,849 kilometers over 70 days, and included 35 city stops. Sushil Reddy, the author who is also an engineer, describes what factors drivers must consider to extend their range. “Lower speeds will generally give a higher range due to the lesser air drag and rolling resistance,” Reddy writes. The feasibility of an EV is also a financial math problem. Within the article, Reddy describes his analysis of cost savings.

Classroom Activities: algebra, efficiency

  • (Mid level) Suppose you are going on a 1,100 mile road trip in an electric car from Los Angeles to Seattle. Your car has a range of 300 miles, but must not let the range drop below 15% of the maximum. 
    • Assuming you leave Los Angeles with a full charge, what is the fewest number of chargers required to drive to Seattle and back?
    • Where should these chargers be (in terms of the number of miles away from Los Angeles)?
    • Write an algebraic expression for the remaining range, y, in terms of miles driven, x.
  • (High level) Suppose that heating the car makes the car 10% less efficient. Write a new algebraic expression. 
    • How much less efficient would the car have to be to require one extra charging station between Los Angeles and Seattle?

—Max Levy

Can You Make a Speedy Delivery?

FiveThirtyEight, January 20, 2023.

Every other week, Zach Wissner-Gross posts two mathematical puzzles as part of FiveThirtyEight’s series “The Riddler”. The second puzzle on January 20 (the “Riddler Classic”) combined geometry and randomness into a problem about how a delivery drone compares to a delivery scooter. In the puzzle, the drone can travel in a straight line to its destination — flying over buildings, cars, and trees — while the scooter must follow city streets. Wissner-Gross asks how much better the drone does at its deliveries than the scooter, requesting the ratio between the average number of deliveries each performs. (In this puzzle, you only have to worry about the distance from the restaurant to deliveries, not distances between two deliveries.) We’ll use this puzzle as an opportunity to build students’ intuition about probability and expected value.

Classroom Activities: probability, geometry

  • (Mid level) Imagine you are at the center of a circle whose radius is 1 mile, and someone randomly chooses a point inside the circle. What is the probability that the point will be less than half a mile away from you? Why?
    • Is it more likely that the point will be less than 1/10 of a mile away, or more than 9/10 of a mile away? Why?
    • Imagine you draw a square anywhere inside the circle. The square’s area is $\pi/4$. What are the chances that the random point will fall inside the square? Why?
    • How do your answers change if you replace the circle with a square whose sides have length 2?
  • (Mid level) The puzzle is about two expected values: The expected distance the drone travels, and the expected distance the scooter travels. An expected value is an average over all possible (random) outcomes, weighted by their probabilities.
    • Imagine playing darts on the diagram below. The smallest circle has radius 1, the middle circle has radius 3, and the large circle has radius 5. Your score is the number labeling the region where your dart lands. What is the expected value of your score? (Assume you’re not very good at darts, and your dart lands randomly anywhere on the diagram.)
Three concentric circles. The smallest circle is marked with a 3, the area between the smallest and the middle one is marked with a 4, and the outer ring is marked with a 6.
Scoring diagram for game of darts. Created by Leila Sloman.
  • Draw a circle with three or four dots randomly placed inside. Measure the distances the drone and the scooter would have to travel to reach the dots from the circle center. Does FiveThirtyEight’s puzzle answer ($4/\pi$) match your measurements? If you know integral calculus, read the full puzzle solution. 

—Leila Sloman

Some more of this month’s math headlines: