Math Digests March 2022

How to play Nerdle, a math-based version of Wordle

Dot Esports, February 28, 2022

It was only a matter of time until someone came up with an even nerdier version of Wordle, the viral word game. Enter: Nerdle. Data scientist Richard Mann created Nerdle for people who enjoy math games, according to writer Sage Datuin. In Nerdle, players guess equations instead of words. Each equation contains digits, operations ($+$, $-$, $\times$, and/or $\div$), and one equals sign. Players must guess equations until they have the right pieces to find the mystery equation. Much like Wordle, you only get 6 guesses, and each guess has to be mathematically correct—or else the game will tell you, “This guess does not compute!” In this article, Datuin highlights the differences between Nerdle and Wordle and shares some tips on how to win.

Classroom activities: Nerdle, combinatorics

(All levels) Play Nerdle individually or as a class. For an easier game, go to the menu and select “mini-Nerdle” to activate a smaller board. Discuss what makes the game hard and what strategies you can use to solve the puzzle in as few guesses as possible.
(Algebra, Statistics) Wordle relies on a fixed dictionary containing 2,315 possible words, but how many different equations are possible in Nerdle? For simplicity, let’s assume that the format will always be a sum or product of two two-digit numbers, and that a three-digit solution is ok, such as 12+34=46 or 12$\times$34=408.
- Consider the spaces on the left side of the equation that contain digits—how many are there, and how many options do we have to fill them?
- Next, how many options do we have for the operation? Now, combine these measures according to the Fundamental Counting Principle.
- Discuss why we don’t need to count the permutations of the digits on the right side of the equals sign.
- (Advanced) How does the calculation change if we allow subtraction? What about division?

—Max Levy

What a Math Party Game Tells Us About Graph Theory

Quanta Magazine, March 24, 2022

Is it possible for every person at a party to shake hands with an odd number of people? Patrick Honner suggests you try it out at your next social gathering. In this article, he explains the solution using the tools of graph theory. The puzzle connects to the concept of subgraphs, which is an open area of research. The latest advance in describing odd subgraphs, Honner writes, came just last year.

Classroom activities: graph theory, even and odd numbers

(All levels) Before students read the article, split them into groups of 5 or 6 and ask them to try to shake hands with an odd number of people. (You can use elbow bumps or verbal greetings instead.)
- Draw graphs representing each group’s handshakes.
- Discuss which groups were able to succeed and why, using the idea of a graph’s “degree sum.”
(Mid level) Watch this 3Blue1Brown video about the three-utilities problem, another popular puzzle in graph theory.
- The three-utilities problem is impossible on a flat surface, but a solution does exist on a coffee mug. Why doesn’t a similar trick work for the shaking hands puzzle?
(Upper level) Complete the exercises at the end of the article (answers are provided).

—Scott Hershberger

Inside the fight to protect your data from quantum computers

Scienceline, March 4, 2022

Every computer you own is in a sense the same under the hood: Desktops, laptops, and phones all run on the same kind of math. Encryption is designed with this in mind. But quantum computers are different, exploiting properties of quantum physics to do their calculations. This allows them to solve certain problems that have been the basis of cryptography until now, such as factoring large numbers efficiently. Quantum computers are hard to build—they require extremely cold metals and other specialized engineering—but if they get big enough, they’ll be able to break much of the encryption that keeps digital communications private. The US National Institute of Standards and Technology is running a competition to find new methods of encryption that are safe against both regular and quantum computers. Most of the finalists rely on mathematical objects known as lattices, as Daniel Leonard writes in Scienceline.

Classroom activities: cryptography, quantum computing, number theory

(All levels) Work in pairs through the lattice-based encryption algorithm described in the graphic in Leonard’s article.
- One person, You, should choose a secret number $S$ and perform Steps 1–3 and the first part of Step 4.
- The other person, Friend, should choose a secret message, $0$ or $1$, and under Step 4 carry out Friend’s Steps 1 and 2 (labeled in blue).
- You should then use Steps 5–6 to uncover the secret message.
- You and Friend exchange roles and repeat the above steps so that each can try both sides of the algorithm.
- You and Friend share the public steps—the first part of Step 4 and Friend’s (blue) Step 2—with another pair of students. Can they determine what secret message Friend sent? If not, what missing piece of information would allow them to find out the message?
(Advanced) Watch a video by minutephysics about how quantum computers break encryption.
- Check the “repeating property” used in Shor’s Algorithm for $g=5$, $N=31$, and $p=3$ by showing that $5$, $5^4$, and $5^7$ all have the same remainder after dividing by $31$.
- Show that for any integers $g$, $N$, and $p$ satisfying the equation $g^p = m \cdot N + 1$ for some integer $m$, the numbers $g^{x+p}$ and $g^x$ have the same remainder after dividing by $N$ (where $x$ is any integer). (Hint: use the fact that $g^{x+p} = g^x \cdot g^p$.)

Related Mathematical Moments: Securing Data in the Quantum Era.

—Tamar Lichter Blanks

Math Is More Than Just Numbers: Celebrate Pi Day a Different Way

Scientific American, March 14, 2022

March 14, or $\pi$ (Pi) Day, has turned $\pi$ into the most famous number. Writing in Scientific American, mathematician Alissa S. Crans takes issue with pi’s fame. “I’m tickled that honoring something mathematical has become a widespread phenomenon,” she writes. “But, at the same time, I’m disappointed that this numerical celebrity seems to be somewhat of an accident.” Crans wants people to understand that math is more than just weird numbers. Math is full of fascinating mysteries and clever solutions to deceptively hard problems. In that spirit, the International Mathematical Union has turned March 14 into the annual International Day of Mathematics. In this article, Crans shares examples of other fascinating facets of math worth celebrating every March.

Classroom activities: pi, cake-cutting, irrational numbers, infinity

(All levels) Explore all the ways that NASA uses $\pi$ to solve its space travel problems.
- (Geometry) Solve this puzzle about a space probe photographing a dwarf planet in our solar system.
- (All levels) Check out NASA’s many other $\pi$-related puzzles.
(Mid level) Crans writes that math can teach you the scientifically perfect way of cutting a cake so that every person gets the same amount, with the fewest cuts. Watch this Numberphile video about the cake-cutting problem to see how this works.
(High level) The decimal representation of $\pi$ contains a non-repeating sequence of digits after its decimal point because it is an irrational number. But Crans notes that $\pi$ is not special in this regard. She writes: “If you asked a genie to choose a number truly at random, the likelihood it would pick an irrational number is 100 percent!” Discuss why this is true. (Hint: it has to do with different sizes of infinity and the fact that the rational numbers have measure zero.)