Math Digests February 2023

The Quest to Find Rectangles in a Square

The New York Times, February 7, 2023.

In December, mathematical physicist John Carlos Baez posted a question about rectangles on the social network Mastodon. The problem was as follows. Take a square and break it into four rectangles, where the rectangles can be of any size and orientation, but must be “similar.” In other words, the rectangles should all have the same proportions, that is, the same length-to-width ratio. Baez asked: how many different solutions are there? Since the original post, mathematicians and math enthusiasts around the world have shared their progress, chipping in with a combination of computer code and mathematical theory. The unofficial team found 11 possible ways the rectangles can be proportioned in a solution, as Siobhan Roberts reports for the New York Times.

Classroom Activities: geometry, ratios, irrational numbers

  • (All levels) Read the original Mastodon post about the problem, where Baez shows three possible ways to break a square into three similarly proportioned rectangles.
    • Draw a square on a piece of paper. Try to divide it into four similar rectangles. Then discuss your ideas with another student. If you both found a solution, did you find the same one, or two different ones? 
  • (High level) Read the first two paragraphs in the beginning of Baez’s blog post on the problem, then take a look at the image with the three squares. (For this activity, it may help to have printouts of this image of the three squares.)
    • Assume that the square has dimensions 1 unit by 1 unit. For the first two pictures, label the lengths of the sides of each rectangle. For a hint, note that Baez describes these rectangles’ proportions in the two bullets under the image.
    • Now, read the third bullet under the image, along with the paragraph that begins “What’s $x$?” Label the lengths of the sides of the rectangles in the third square, using “$x$” the way that Baez does in his description.
    • For a challenge, read through the algebraic explanation of $x$, up through the line $\rho^3 = \rho + 1$. Discuss: Does it seem surprising to you that the height-to-width ratio of the rectangle is an irrational number? Why or why not?

—Tamar Lichter Blanks

The Math Behind the Medals: Professor Ken Ono Is Helping Virginia Revolutionize Swimming Performance

Swimming World Magazine, January 31, 2023.

In 2021, when members of the University of Virginia swim team attempted to qualify for the Beijing Olympics, they had a secret weapon: mathematician Ken Ono. They needed more than just muscles to swim faster than they ever had before. By combining sensors like accelerometers with mathematical analysis, Ono carefully studied the swimmers’ form and suggested tweaks to improve their speed. One swimmer named Paige had ranked seventh in a 200-meter contest and eleventh in the 400-meter. Only the top six swimmers could advance, so she considered only focusing on the 200-meter. But Ono’s data revealed a different story: “I had been telling [UVA’s head coach] and Paige, you really want to be focusing on the 400,” Ono told Swimming World writer Mathew De George. “Our speculation was that Paige would have almost a near lock to make the Olympic team.” That prediction came true. This article describes Ono’s analysis and how he found this niche. 

Classroom Activities: movement analysis,

  • (Mid level) Install an accelerometer app on your cell phone (individually or in groups) with a “record” function.
    • The forces graphed on this app should have three components: $x$, $y$, and $z$. Which dimensions/directions of movement does each variable correspond to? (up and down, side to side, or forward and back)
    • Start a recording on the accelerometer. Place your phone in your hand and move it from side to side as if waving hello. The accelerometer will produce a chart showing force versus time. Based on this chart, how long does one side-to-side movement take?
    • Predict how the accelerometer graphs for $x$, $y$, and $z$ will look for the following movements and sketch them. Then perform the movement and compare the graph to your prediction.
      • A full circle at constant speed (as if wiping a window)
      • A full circle at constant speed (as if wiping a table)
      • A zig zag motion facing you

—Max Levy

What’s Going On in This Graph?

The New York Times, February 15, 2023.

On February 7, the basketball player LeBron James scored the 38,388th point of his career. The achievement made headlines, as it meant James had broken the record for most career points in the NBA, previously held by Kareem Abdul-Jabbar. The Times’ analysis of the event included a graph showing how the 250 top scorers in the NBA accumulated points throughout their career, highlighting the line representing James. Two days later, the Learning Network posted the graph as part of their “What’s Going On in This Graph?” series, which asks students to analyze and interpret a graph and post their thoughts for others to consider.

Classroom Activities: data analysis, point-slope form

  • (All levels) Answer the four questions asked by the Learning Network in activity 1. When you’re done, find a partner and share your answers. What did your partner pick up on that you didn’t notice? What did your answers have in common?
  • (Algebra) Read this online lesson on point-slope form. Now, use the graph to apply it to real data. By eyeballing two data points from James’ scoring graph, estimate the formula of the line. Repeat this with 3 different choices of points. Do the same for Abdul-Jabbar.
    • Print out the graph, and draw the lines whose formulas you estimated. How well do they match the data? Which choices of points created a line that best matched the data?
    • James is 38 years old, four years younger than Abdul-Jabbar was when he retired. Using your formula, estimate how many points James will have if he plays until he’s 42. How accurate do you think your guess is?
  • (Algebra) Create your own graph with data you collect from your daily life. For example, you might use your phone to track how many steps you take each day, and graph your total steps for the week. After one week, repeat the activity above, finding a line and predicting what the total would be if you continued for three more days. After three more days, check how good your estimates were.

—Leila Sloman

Punxsutawney Phil’s Groundhog Day prediction accuracy rate calculated

Cleveland19 News, February 2, 2023.

Every year, a groundhog in Pennsylvania predicts the weather. Or at least, he tries to. If Phil sees his shadow on Groundhog Day, it is believed that there will be six more weeks of winter. The tradition dates back to the 1800s, and in 2023, Phil predicted that winter would stick around for another six weeks. “To verify whether or not his predictions have been accurate, we look at temperatures across the country during the months of February and March,” writes Erika Paige, in an article for Cleveland 19 News. Paige questions the accuracy of Phil’s predictions and describes her process for conducting the analysis. Since records have been kept, Phil has failed to see his shadow only 20 times. Over the last 30 years, Phil’s success rate is only around 37%. 

Classroom Activities: data analysis, spreadsheets  

  • (Mid level) Transfer the table of data from the article into a spreadsheet, such as Microsoft Excel. Use spreadsheet functions to do or answer the following.
    • In the last 20 years, how many times did Phil see his shadow? (Use equation functions, do not count manually)
    • What percent of Phil’s predictions of winter were correct in the last 20 years?
    • What percent of Phil’s “spring” predictions were correct, in the last 20 years?
    • Use the software’s Conditional Formatting options to make all “verified” predictions green.
    • What was the longest streak of verified predictions in the last 30 years?
  • (High level) The article determines a “true” winter or spring based on whether the actual temperatures are warmer or colder than average for February and March. As a class, talk about whether you believe this is a fair classification or not. If not, then how would you change it? Do you expect that this change should increase Phil’s calculated accuracy or decrease it?

—Max Levy

Even the Smartest Mathematicians Can’t Solve the Collatz Conjecture

HowStuffWorks, February 14, 2023.

Pick a number, any number. If it’s odd, multiply it by 3 and add 1. If it’s even, divide it by 2. Apply this recipe again and again until you get 1. The Collatz conjecture states that this is guaranteed to happen, no matter what number you pick. But as Jesslyn Shields writes for HowStuffWorks, no one has actually been able to prove this — they’ve only been able to check all the numbers that have 19 or fewer digits. There are still infinitely many numbers for which the conjecture is still unknown.

Classroom Activities: sequences, proofs, programming

  • (All levels) As Shields writes, the sequence of numbers you get by applying the Collatz recipe step-by-step is called the “Hailstone sequence”. Find the Hailstone sequences that start with 1, 3, 20, and 13.
  • (Mid level) Imagine a simplified version of the recipe: At each step, multiply by 3 and add 1, whether the number is odd or even. What will happen if you apply this recipe over and over?
    • What if you divide by 2 at each step? What will happen if you apply that recipe over and over?
  • (High level) Change the Collatz recipe so that if you get an odd number, you simply add 1. Re-calculate the sequences from the last activity with this change. Prove that with this recipe, you’ll always end up with 1.
    • Now tweak the recipe so that if you get an odd number, you multiply by 2 and then add 1. Re-calculate the sequences from the last activity with this change. What’s different about this sequence? What do you think will happen if you follow the sequence forever?
  • (Programming) Write a computer program that applies the Collatz recipe to every integer between 1 and 100, and outputs the number of steps it took to reach 1. Describe the results. Do they make you more or less convinced that the conjecture is true?
    • Shields writes: “[Terence Tao]’s results point to a new method for approaching the problem and note how rare it would be for a number to diverge from the Collatz rule. Rare, but not necessarily nonexistent.” Is this consistent with the data from your program?

—Leila Sloman

Some more of this month’s math headlines: