**Probability**(Grades 5 through 12)

**Crazy Dice**(Grades 5 through 12)

**Evening the Odds**(Grades 7 through 12)

**Crazier Dice**(Grades 7 through 12)

Probability is a way to quantify the likelihood of an event occurring. Students explore this concept through guided interactive experiments and challenges. Some prior exposure to basic probability and familiarity with adding fractions is assumed for these programs.

Once students find the probability of rolling a given sum with a pair of standard dice, they are challenged with finding a different way to number their dice to get the same probabilities. Crazy!

The probability of rolling different sums with a standard pair of dice depends upon the sum in question. This activity guides students to find ways to renumber their dice so every sum appears with the same probability. But then, what sums are possible to fit into this scheme? How many different sums can be rolled with equal likelihood? The answers to these questions and more are found in this activity.

If renumbering a standard pair of six-sided dice seems crazy, you might want to sit down for this activity. Students work together first to understand the probabilities encountered when rolling eight-sided dice, and then to find alternative ways to number these octahedral dice without changing those probabilities.

Madeline Chun, Mathematics and the Marimba

In her project, Madeline uses the musical instrument known as the marimba to demonstrate something profound about music: musical harmony is intimately connected to the ratios of small whole numbers, a discovery that Pythagoras and his followers are said to have made 2500 years ago. The judges were impressed with the clarity and elegance of Madeline’s explanations and demonstrations.

Click here to watch Madeline’s video.

Jibran Iqbal Shah, @creative_math_

Jibran has an Instagram site, @creative_math, where he introduces high school students to topics normally reserved for undergraduate math majors. The judges admired the attractiveness of his posts and how skillfully he explains advanced mathematical ideas, always with a friendly voice.

Click here to read Jibran’s posts.

Alyssa Choi, Tic-Tac-Toe

Alyssa wrote a paper that explains the optimal strategy for tic-tac-toe and also explores variants of the game on a 4×4 grid and in a 3x3x3 cubical lattice in three dimensions. The judges appreciated the clarity and originality of Alyssa’s investigations.

Click here to read Alyssa’s paper.

Cole Leis, On the Importance of Humanizing Math Communication

Cole’s project was an essay about “a disconnect between the way math is typically portrayed and the way that it is actually done.” Math textbooks and papers usually try to look flawless, but anyone who does math knows how messy and confused it can be when you’re actually doing it. Cole interviewed several mathematicians about this disconnect and produced a cogent argument that math communication would be more effective if we humanized it by showing our mistakes as well as our insights.

Click here to read Cole’s essay

Ian Liu, The Golden Phi-ano Song

For his project, Ian constructed a song out of the digits of the irrational number known as the golden ratio (also known as the divine proportion because of its beauty). All the judges found the project intriguing as an exploration and thought the song itself was surprisingly pretty.

Click here to listen to Ian’s song.

Madeleine Farr, A 3D Valerie Thomas, But Not How You'd Think

Madeleine created a portrait of the mathematician Valerie Thomas. The judges were impressed with this project’s inventiveness: it uses nothing but the characters “3D”—artfully placed—to create a powerfully evocative portrait of a mathematician known for her work on 3D satellite imaging.

Helena Soares, This Fundamental Theorem - A Found Poem

Helena wrote a “found poem” – a poem where every word (except for “you” and “us”) is from another source, in this case the MathWorld article by Eric Weisstein on “Fundamental Theorems of Calculus.” Helena rearranged the words to create something totally unexpected, an understated little poem that took the judges’ breath away.

Click here to read Helena’s poem.

Shreya Mogulothu and Amy Wang, Combinatorial Proofs... and Basketball

Shreya and Amy teamed up to bring a combinatorial identity to life. Using basketball as a mathematical playground, they showed two approaches to calculating how many ways a player could make or miss *n* shots, and thereby neatly led the viewer through a proof that the sum of the binomial coefficients “*n* choose *k*” from *k* = 0 to *n* equals 2^*n*. The video is goofy and fun: Amy writes (and playfully trash-talks) while Shreya shoots and, before you know it, they’ve derived the identity for the sum of the binomial coefficients. The judges enjoyed the simple but clever videography, the banter between the girls, and their neat explanations of the math.

Click here to watch the video

Julia Schanen, Math Person

Julia Schanan’s entry for the Strogatz Prize was a free-verse poem titled “Math Person.” The judges were moved by the poem’s artistry and emotional power, its depth and raw honesty, its brilliant use of language, and its eye for the unexpected but telling detail. “Math Person” conveys – in ways both beautiful and haunting – the isolation Julia felt as one of the only girls in the American Math Competition 10th grade and, more profoundly, the intellectual isolation she still feels every day as someone who loves math deeply yet lacks a friend with whom to share it.

*Mom offers to stop by Panera as a treat for all the painful math that I’ve just endured.*

*Except it wasn’t painful.*

*I’m someone who sat through the slow-drip of middle school math, bored and daydreaming,*

*not seeing what it was all for, wishing – but never working up the guts to push – for more.*

*Not until now.*

*Now, I don’t want Panera.*

*I don’t want to be patted on the shoulder and misunderstood.*

*I want to go back into that auditorium and finish the exam and talk about it all night.*

The judges felt that their own words were inadequate to summarize Julia’s achievement in writing “Math Person.” Let us simply say, read her poem and experience it for yourself.

Click here to read Julia’s poems.

Apoorva Panidapu, Gems in STEM

Apoorva Panidapu is a 16-year-old mathematics student, artist, and advocate for youth and gender minorities in STEAM. She writes a blog called “Gems in STEM” and frequently posts the essays on Cantor’s Paradise, the #1 math site on Medium.com. She sees her blog as “a place to learn about math topics in an accessible, light-hearted manner. I assume no more than basic math knowledge and include fun tidbits for learners of all experience levels. For both my own fun and for readers, I weave in pop culture, pick-up lines, and over-the-top stories to let people into the fantastical world of math, and to show them that anyone can enjoy anything.”

The judges were very impressed with Apoorva’s joyful, elegantly written blog posts on a wide range of math topics, from the liar’s paradox and partitions to tessellations and fractals. Combining clear explanations with an appealing layout and well-chosen graphics, Gems in STEM is itself a gem. The judging panel loved the wide range of Apoorva’s blog posts. They touch on history, etymology, and puzzles, and make connections to everything from art and architecture to science and nature. Apoorva’s uplifting message is that math is everywhere and approachable by anyone from any background.

Click here to read Apoorva’s posts.

Shirley Zhu, Combinatorics: Sticks and Stones

Shirley’s animated video is filled with jokes and pop-culture references, intended to appeal to younger audiences (though the audience of older judges enjoyed it very much too). Her goal, she says in her essay, is to demystify “mathematics with humor, fun animations, and real-world scenarios.” In her video, Shirley illustrates a technique in combinatorics known as Sticks and Stones (or Stars and Bars) by telling a story about Grandpa Bob’s dilemma: He wants to know how many different ways there are for him to distribute 10 one-dollar bills among his three granddaughters. With excellent pedagogy (including falling into a tempting mathematical trap and then showing how to get out of it), the video shows the viewer how to count all ways Grandpa Bob can solve his problem. The video is fun and fast-paced, but spends just the right amount of time on the harder ideas and leavens them with colorful graphics and word choice.

Click here to watch Shirley’s video

Catherine Stanton, Understanding Set Theory

Catherine’s video was inspired by her struggle to understand why the empty set is a subset of every set. As she wrote in her essay for this contest, “I’ve never been one to accept or memorize a mathematical concept unless it makes complete sense to me, so I decided to go beyond sheer numbers and create an analogy. I realized that the sets on my page could be related to my classes at school. As the elements of a mathematical set are numbers, the elements of a class are students. When some are absent, only a “subset” of the class exists. And when all are absent, a subset of the class still exists since the class period still technically occurs. This analogy wasn’t something I wanted to keep to myself, and so it became the basis of my entry to the Strogatz Prize competition!”

The judges appreciated Catherine’s lucid explanations and accompanying graphics. Set theory is often regarded as one of the most abstract parts of math, but in Catherine’s hands, it becomes tangible and vivid. Her classroom analogy should appeal to any student. Beyond that, Catherine gives the viewer a glimpse of the universality of set theory and a taste of its applications in computer science.

Click here to watch Catherine’s video

Smayan Khanna, 4 Colour Map Theorem

In this video about a famous theorem in graph theory and topology, Smayan beautifully explains why every planar map can be colored with just four colors, in such a way that neighboring countries always have different colors. The video is gentle and clear, and enlivened by plentiful graphics, neatly synchronized to Smayan’s narration. He gives the viewer helpful intuition about how inequalities (for vertices, edges, and faces of planar graphs) can be used to cut a seemingly infinite problem down to a finite number of cases, which can then be checked by computer. The judges felt that this was a very crisp presentation of a deep result in advanced mathematics, explained in a way that anyone can grasp and enjoy.

Click here to watch Smayan’s video

Math, Music, and the Moral Imagination

Jazz saxophonist and mathematician Marcus G. Miller will share a reflection on how math and music can make us whole.

**Marcus G. Miller** is a musician and mathematician living in New York City.

Logic Puzzle

John Urschel, current MIT math PhD candidate and former NFL pro, shares his favorite logic puzzle.

**John Urschel **played professional football for the Baltimore Ravens from 2014 to 2017 before retiring to focus on his career in mathematics. He is currently a PhD candidate at MIT, where he studies spectral graph theory, numerical linear algebra, and machine learning.

Brainteaser Kahoot

Get ready to take part in a whacky and zany brainteaser Kahoot with Steve Sherman. This is a quiz that will tickle your brain-strings and challenge your thinking skills. Some of the brainteasers will be easy while others will make you think. Do you have what it takes to be our brainteaser champion?

**Materials needed:**

- Smart phone, tablet, or personal computer with internet access

**Steve Sherman** is the Chief Imagination Officer and Executive Daydreamer of Living Maths.

The Doomsday Rule

When visiting MoMath, the late, great John Horton Conway impressed people by instantly telling them the day of the week that any given date fell on or will fall on. And you can do it too! What date of the week will your birthday fall on next year? Or your anniversary? Or favorite holiday? When’s the next Friday the 13th? What day of the week were you born on? Tune in for a presentation by MoMath Puzzle Master Peter Winkler on the marvelous “Doomsday rule.” It’s easy and fun, and with a little practice you’ll be able to duplicate Conway’s feat. It’s a great trick, and it’s cleverly designed so that you can remember it and wheel it out whenever it’s needed.

**Materials needed:**

- Paper
- Pencil

**Peter Winkler** is the current MoMath Distinguished Chair for the Public Dissemination of Mathematics and Professor of Mathematics and Computer Science at Dartmouth College.

Moving in Circles

Join Karl Schaffer as we play with several surprising ways of moving our limbs in circles. Apply these actions to create movement sequences with the ultimate mathematical prop — an ordinary sheet of paper. Then, learn how it connects to the curious algebra of quaternions!

**Materials needed:**

- Several sheets of 8.5″ by 11″ printer paper
- 5′ by 5′ area in which to move (non-carpeted area preferred)

**Karl Schaffer** is a dancer, choreographer, mathematician, and math professor at De Anza College.

Conway's Rational Tangles

What can the fluidity of topology and knots have to do with the rigidity of operations on fractions? Join Alex Kontorovich to learn the fascinating connection discovered by the late mathematician John Conway.

**Materials needed:**

- 2 strings of different colors (wires or shoelaces are also okay)
- Surface to lay the strings on (e.g. floor or table is fine)
- Optional materials: paper and pencil

**Alex Kontorovich** is MoMath Dean of Academic Content and math professor at Rutgers.

Folding Fun!

Origami expert Wendy Zeichner will teach one or two origami action models that will be folded from regular printer paper. There’s math in every fold!

**Materials needed:**

- Several sheets of 8.5″ x 11″ printer paper
- Pen or pencil
- Scissors

**Wendy Zeichner** is an origami expert and president of OrigamiUSA.

Math in Motion: Build Your Own Paper Spinner

Turn two pieces of heavy paper and some tape into a spinning top, and explore the geometry behind it! Join Yana Mohanty, Ph.D., a mathematician and inventor of Geometiles®, as she guides you through this fun STEM activity. You will be provided with a printable template and shown how to transform it into your spinner. Once you master the simple construction, you may decorate your spinner with your own design.

** Materials needed** (metric equivalents in brackets)**:**

- 2 sheets of 8.5” x 11” (size A4) card stock paper, ideally in 2 different colors; manila folders cut to 8.5” x 11” will also work
- Scissors
- Tape
- Optional: This project can also be built from Geometiles® Mini Set 2, available from
*Additions*, the online shop at MoMath.

**Yana Mohanty** is a math educator, mathematician, and the creator of Geometiles®.

One Step Back, One Step Forward

If repeated flips of a coin determined which step you took, would you get anywhere? Step right in (bring your own coin) and find out!

**Materials needed:**

- 1 coin
- Paper
- Pencil

**Ralph Pantozzi** is the winner of the 2014 Rosenthal Prize and a Presidential Awardee in math teaching.

Numbers Meet Shapes in 3D Geometry

Let’s discover the magic of Euler’s Polyhedral Formula while creating structures out of toothpicks and marshmallows.

**Materials needed:**

- 1 box of traditional rounded toothpicks with points at both ends
- 1 bag of mini marshmallows
- Paper
- Pencil

**Bruce Bayly** is a math professor at the University of Arizona and bus driver for the Arizona Mathematics Road Show.

Engineering with Paper

Join us online for a math-and-paper engineering adventure! Godwyn Morris, Director of Dazzling Discoveries STEM Education Center, will demonstrate some Engineering with Paper challenges. Together we will explore proportion, ratio, and scale as Godwyn shows you how to create structures, furniture, and characters from simple supplies.

**Materials needed:**

- Printer paper
- Tape
- Scissors

**Godwyn Morris** is the Director of Dazzling Discoveries STEM Education Center.

Mathemagics!

Dr. Arthur Benjamin will amaze you with some mathematical magic, and then teach you how to do it. Bring a calculator!

**Materials needed:**

- Calculator

**Art Benjamin** is an award-winning popular entertainer, mathemagician, and math professor at Harvey Mudd College.

How to Fold Things into Thirds, Thirty-Thirds, and Three-Hundred-and-Thirty-Three-"irds"

Bring along something bendy — a tie, a shoelace, a piece of string — and let’s have fun exploring some of the curious mathematics of folding.

**Materials needed:**

- Something bendy, such as a tie, shoelace, or piece of string

**James Tanton** is a global educator, math popularizer, and mathematician.

Can Math Help You Juggle?

Educator and entertainer John Chase will show you the powerful connections between mathematics and juggling. Math modeling has given jugglers all kinds of new patterns to juggle, and we invite you to come see what mathematics can do. Bring three juggling objects so you can join the fun!

**Materials needed: **

- 3 juggling objects

**John Chase** is a mathematical juggler and math educator.

A Four-Card Mathematical Magic Trick

Manjul Bhargava will demonstrate an interactive magic trick that exhibits how one can create surprising complexity from extreme simplicity. Viewers are encouraged to participate from home!

**Materials needed:**

- 4 playing cards

**Manjul Bhargava** is the MoMath Inaugural Distinguished Visiting Professor for the Public Dissemination of Mathematics, math professor at Princeton University, and Fields Medalist.

Activities from the Julia Robinson Mathematics Festival

Come and enjoy some non-competitive explorations in mathematics.

**Mark Saul** is the Senior Scientist at the Julia Robinson Mathematics Festival.

Let the Math Fun Begin!

Cindy Lawrence, MoMath Executive Director, and Tim Nissen, MoMath Associate Director, welcome all to the fifth annual NYC Math Festival, featuring hours of mathematical fun and entertainment. Join presenters from all over the world to share an afternoon of lively, engaging activities.

James Tanton, Chair of MoMath’s Advisory Council, kicks the Festival off by sharing the highly intriguing “International Math Salute.” Can you figure out how it works?

NYC Math Festival Pre-Show

The Festival site opens at 1:30 pm EDT (New York). Enter at any time between 1:30 pm and 2:00 pm to view some favorite MoMath pix from years gone by. Activities begin promptly at 2:00 pm EDT (New York).

See our selection of Geometiles® and browse for books by Festival presenters Peter Winkler and Art Benjamin.

A MoMath retail specialist will be on hand to answer questions and offer expert shopping advice for all your mathematical gift needs.

Stop by to view a selection of Rubik’s Cubes, browse for Christopher Danielson’s delightful books, *Which One Doesn’t Belong? *and *How Many?*, and learn more about the film *The Man Who Knew Infinity*, for which Manjul Bhargava served as technical advisor to ensure the accuracy of the math featured in the film.

A MoMath retail specialist will be on hand to answer questions and offer expert shopping advice for all your mathematical gift needs.

Check out MoMath’s selection of colorful origami paper, Karl Schaffer’s book *Math Dance*, and other great items from *Additions, *the online shop at MoMath.

A MoMath retail specialist will be on hand to answer questions and offer expert shopping advice for all your mathematical gift needs.

Math Meets Art: Rubik's Cube Mosaics

Watch as Lauren Rose builds a Rubik’s Cube mosaic, and try to figure out what the picture will be. Math meets art in this creative application of the popular Rubik’s Cube.

**Lauren Rose **is a mathematician and math professor at Bard College.

Rubik's Cube With Feet

Daniel Rose-Levine will demonstrate how he solves the Rubik’s Cube with his feet in under 20 seconds.

**Daniel Rose-Levine** is the former Rubik’s-Cube-with-feet world record holder.

Mime-matics

How can the invisible world of mime explore mathematical ideas? Join Tim Chartier as he uses mime to investigate weight, magnitude of force, and projective motion with the art of mime. You’ll also learn how to make an infinite chocolate bar. (The candy bar is a real prop but eating is pantomimed so enjoyment is calorie-free.)

** Tim Chartier** is a mathematical mime performer and math professor at Davidson College. He has performed throughout the world and has been trained in several mime schools, including master classes with the legendary Marcel Marceau.

Finding and Creating Symmetry

Symmetry is all around us. We see symmetry in our bodies, car wheels, fences, fabric patterns, the MoMath logo, and many other objects! We will learn about different types of symmetry and have fun creating symmetric art using common objects.

**David Reimann** is an Albion College math and computer science professor and artist who uses symmetry in his work.

Fun with Mathematical Definitions

Is an elevator a vehicle? Is a hot dog a sandwich? Is a heart a shape? Is an emoji a word? The answers to these questions depend on your definitions of vehicle, sandwich, shape, and word. Precise definitions are essential tools of mathematics, but few definitions start out as precise as they’ll need to be later on. Come play with the boundary between precision and ambiguity in this fast-paced participatory session.

**Christopher Danielson** is an award-winning author and math educator.

Math Raps and the Math Behind Them

Join Mr. A. as he shares one of his many Math Raps, discusses how he got started rapping about math, and takes you through some of the mathematical ideas and references in the rap.

**Mike Andrejkovics** is a high school math teacher from Long Island, NY who creates and performs raps about mathematics based on popular hip-hop tracks.

Christopher Danielson is an award-winning author and math educator.

Cindy Lawrence is the Executive Director and CEO of the National Museum of Mathematics.

Tim Nissen is the Associate Director of the National Museum of Mathematics.

Steve Sherman is the Chief Imagination Officer and Executive Daydreamer, Living Maths.

Karl Schaffer is a dancer, choreographer, mathematician, and math professor at De Anza College.

Alex Kontorovich is MoMath Dean of Academic Content and math professor at Rutgers.

Bruce Bayly is a singer, violinist, and math professor at the University of Arizona.

Daniel Rose-Levine is the former Rubik’s-cube-with-feet world record holder.

Godwyn Morris is the Director of Dazzling Discoveries STEM Education Center.

James Tanton is a global educator, math popularizer, and research mathematician.

John Chase is a mathematical juggler and math educator.

Lauren Rose is a dancer, choreographer, mathematician, and math professor at De Anza College.

Mark Saul is the Executive Director of the Julia Robinson Mathematics Festival.

Origami expert Wendy Zeichner is the Executive Director of Origami USA.

Ralph Pantozzi is the winner of the 2014 Rosenthal Prize and a math educator.

Yana Mohanty is a math educator, mathematician, and the creator of Geometiles®.

Carolyn Figuereo is the retail manager of the National Museum of Mathematics.

Tim Chartier is a mathematical mime performer, math professor at Davidson College.

International Math Salute

Details TK

Rohan Jha

“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.

Kyna Airriess

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.”

Sarah Thau

“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

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.

Yvonne Hong

“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

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 1^{2 }+ 2^{2 }+ … + n^{2} 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!

Katarina Cheng

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 E. Alsamraee

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.