1,729 is the “taxicab number”; the smallest number that is the sum of two cubes in two different ways: 1000+729 = 1728+1
17,290 is 10 times the taxicab number.
36,100 is the sum of the first 19 cubes.
50,021 is a prime number, meaning that it has no factors except one and itself.
109,376 is an automorphic number; if you multiply it by itself any number of times, the same six digits remain at the end.
“The purpose of Math Musings, the magazine I started in high school,” wrote Rohan Jha, “was to show that math is everywhere, yet many times we are not aware of it. It is behind some of the music we play, or how nature uses it for its own optimal benefit, or it could be behind a fancy card trick, or math could help us reduce the ubiquitously observed annoyance of traffic jams during peak hours.” The magazine tries to humanize and enliven math in various ways: by telling anecdotes about famous mathematicians; by challenging fellow students with fun puzzles; or by leading them some deeper ideas, such as a lily pad puzzle that leads to the notion of backward recursion in finance. With clear illustrations and step-by-step instructions for magic tricks and other activities, Rohan attempts to make math fun for everyone… and succeeds admirably.
The project submitted by Kyna Airriess is a “zine” based on a quote from A Mathematician’s Lament, a polemical essay by high school teacher Paul Lockhart. “There is nothing as dreamy and poetic, as radical, subversive, and psychedelic, as mathematics,” wrote Lockhart. Reading Lockhart’s essay, says Kyna, “contributed to my own conversion from ardent math-hater to aspiring mathematician; I’d never heard someone describe math, the subject of unfeeling calculations, with words like ‘poetic’ and ‘radical.’ It was a long time before I began to see these traits for myself, but today I self-identify as a math nerd, and I want to study math in college.”
In the zine, each of Lockhart’s memorable adjectives—dreamy, poetic, subversive, and psychedelic—is illustrated and connected to math ideas, using symbols, history, color, and imagery. The judges were impressed by the passionate energy conveyed by the zine’s words and design. The overall effect achieves what Kyna intended: to embody “what those of us who love math want the world to understand. It isn’t about cold calculations at all— it’s a field full of creativity and beauty, and it is just as infused with humanity as any other.”
“Limericks and poetry are not a typical way to convey information about math,” admits Sarah Thau, “but I think it makes it more palatable than learning functions by rote. Who doesn’t love a limerick?” So Sarah created a series of short rhyming poems to list some basic properties of linear, quadratic, trigonometric, polynomial, rational, and other types of functions encountered in algebra and precalculus, and illustrated the pages with examples.
The judges were tickled by the playfulness of this entry. Limericks are a lighthearted form of poetry in which creativity comes from working within constraints and overcoming them delightfully—and much the same can be said of math! Indeed, as Sarah wrote, “I love math and am always trying to solve problems but this was a new type of problem to tackle. One that didn’t need any algebra or modeling. Each poem became a problem to solve as I tried to figure out words to make each function type’s properties rhyme neatly.” The poems illuminate the distinctive properties of the various kinds of functions, and draw readers in through a unique, creative, and memorable way of communicating mathematical ideas.
Jonah Yoshida’s project is a pencil-and-paper infographic on graph theory. He says “I conceived of the idea when reading about how Arthur Cayley used trees to represent structures of hydrocarbons with n carbon atoms and 2n+2 hydrogen atoms. The entire structure imitates one of these hydrocarbons, ethane (n=2), and a unique application of graph theory is included inside each atom. I divided the page into two sections so that the hydrogens bonded to the left carbon contain puzzles and fun applications of graph theory, while the ones bonded to the right hydrogen focus more on direct applications, much like our brains’ left and right hemispheres.” For example, the Four Color Theorem (a fun application of graph theory to coloring maps and an longstanding research question) appears on the left, while the right side includes applications of graphs to computer science (neural networks and spanning trees) and electrical engineering (circuit diagrams).
The judges appreciated the ingenious design concept of this graphic, which underscores the universality and interdisciplinary spirit of graph theory. The words and imagery combine history, math, chemistry, and psychology, and the questions in the small text boxes invite the reader to do some research of their own.
“My math communication project, Infinity Universe,” wrote Yvonne Hong, “is an illustrative yet mathematical depiction of the world in which we live. Every inanimate object illustrated represents a simple, yet ubiquitous concept in math: upon closer inspection, the monochromatic tree is a fractal Pythagoras tree, the galaxy in the background is constructed using the Fibonacci sequence, and the planet and comet are both different variations of the Apollonian gasket. Infinity Universe promotes the universality of math communication through an abstraction of objects and phenomena that people all around our world are familiar with.”
Carefully executed with great attention to detail, the painting submitted by Yvonne drew the judges in with its vibrant colors and hypnotic patterns. Moreover, the theme of infinity pervades the painting, just as it does in all of mathematics. But here, the suggestion of the infinite is magical and otherworldly rather than scientific and literal, and so may appeal to audiences not normally attracted to math.
Zoe Markman created a visual proof of the “sum of squares formula” by cleverly using three wooden 3-D pyramids that fit together. Each pyramid consisted of a total of 12 + 22 + … + n2 identical wooden cubes; thus, its volume visually represented the sum of the squares of all the whole numbers from 1 to n. To find a formula for this sum of squares, Zoe manipulated and rearranged the three pyramids to form a rectangular prism, whose volume could then be easily calculated to obtain the desired formula for the sum of squares.
The judges agreed with Zoe that this sort of visual, hands-on manipulative “provides a deeper understanding of math than that provided by a written project. Since you can observe, hold, and manipulate the pyramids (even more so in person), the audience is able to understand why the formula works rather than just taking it at face value and accepting that it was true arbitrarily. Second, the presentation could be understood even by people without a significant knowledge of math. It put what looks like an intimidating problem in terms that are easily digestible.” Zoe even tested the presentation on friends who said they didn’t like math. That’s a good practice in any form of communication. Overall, this project is modest but extremely well done and produces a very pleasurable “Aha!” moment for many viewers; indeed, it led one of the judges to understand the “sum of squares formula” in a whole new way!
To express the universality of math, Katarina Cheng translated it into another universal language: dance. “Just as dance exists as a part of many cultures around the globe to express abstract ideas and emotions through movement, mathematics defies cultural lines to express abstract ideas through structures and forms on the page,” she wrote in her project description. Her video “Dancing the Dihedral Group” sought, through dance, “to represent the visual symmetries, primarily those of a square,” and, through words, “how they translated into algebra, primarily the group D8 .”
The judges commend Katarina for the elegance of her communication in the video. Especially notable was the esthetic of minimalism — in how the video is shot, and the choice of clothing, background, and colors — all of which mesh perfectly with the minimal esthetic of group theory. The integration of the math graphics with the dance moves was also carried out gracefully. Although others in the past have recognized the similarities between math and dance, few have conveyed that analogy with such finesse in the execution. The dancing and music were artfully minimal too. The overall effect is to reinforce the central idea of beauty in simplicity.
Hamza Alsamraee loves Instagram – and he also loves math. But when he noticed that very few math pages existed on Instagram, he sought to change that by starting @daily_math, a page dedicated to intriguing problems and ideas about algebra, geometry, calculus, number theory, and other parts of math. “With high-quality educational posts,” he says, “I hoped to build an Instagram community centered around a shared passion for math.”
The judges were impressed with the creativity of Hamza’s entry, expressed through its skillful use of visuals, history, and puzzles, all presented in attractive ways. His explanations of mathematical concepts are clear and insightful, and he is very interactive with his followers, even inviting them to post. The judges also commend him on his growth as a creator and communicator. His Instagram page has evolved from a focus on tricky integrals in the early days to doing more accessible problems now, and the visual presentation has evolved in tandem. With his engaging design choices, which foster clear communication, he is making increasingly good use of the strengths of the Instagram medium.