Art and math feel like opposites at first; one uses imagination and creativity, and the other relies on logic and rules. However, math exists behind so much of the beauty we see, especially in art.

One of the most straightforward examples is origami. Folding paper into cranes or stars from sticky notes might seem purely a way to pass time in class through art, but it is based on angles, symmetry, and geometry through algorithms that turn shapes into foldable designs. Through a career lense, many engineers use origami to design solar panels and medical devices, simply from the folding that we learn for fun. 

Symmetry is another place math shows up in art through tile patterns, traditional mandala-style designs, and portraits. Artists often use the transformation skills from graphing, like reflection, rotation, and translation to bring balance to their work. The shifting up and down units is key to the work of many artists. For example, Maurits Cornelis Escher (M.C. Escher), a pioneer in art and mathematics, created mind-bending patterns that were inspired entirely by geometry and tessellations.

We also can’t forget about the Fibonacci Sequence and Golden Ratio; two patterns that appear in shells, flowers, and pinecones. Artists often use the same ratios to design layouts and compositions for the visual satisfaction of their audience. 

In the digital world, math is the backbone of creativity through animations, visual effects, and graphic design. The curves, vectors, and equations bring characters and images from fan-favorite cartoons to life. Behind every beautiful motion graphic and 3D model is the world of mathematics. 

From the doodles and sketches in notebooks to folded stars and hearts, math makes it all work. Art and math go hand in hand, and often, combining them is where real magic happens. 

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This past summer, I had the opportunity to teach math to seventh graders through a month-long internship. The students came from under-resourced, low-income, and first-generation families, where they could not get help from tutors or educated parents if they didn’t understand a concept. 

Students were exposed to ratios, proportions, time, and other math skills used in everyday life. The concepts opened up new ways of thinking that the students could carry with them for the rest of their lives. For example, being able to read the clock accurately or solve a problem on ratios gave them confidence that math is a subject they can love, not run away from. 

One of the best parts of the experience was watching the students’ attitudes toward math evolve over the four weeks. I remember walking in the classroom on the first day, seeing half of the class with heads on their desks, and a few of them talking amongst themselves on how they would “fail the class from being bad at math.” After a few weeks, those same students ran into the room excited for groupwork activities, projects, and games, where they could solve problems with pride and confidence in themselves. Even the smallest of achievements, like correctly solving a ratio on the board or explaining a solution to a peer, created moments of excitement in the classroom.

I received valuable skills too, like patience when I explained the concepts in different ways until it clicked with each and every student. It taught me how important encouragement is, especially when students may not always have access to consistent resources or support.

By the end of the internship, I could see how much the students had grown. They were more confident, willing to try problems on their own, and genuinely interested in learning something new. Watching them improve and gain confidence made the experience so rewarding. It reminded me that teaching isn’t just about passing down knowledge, but more about helping people believe in themselves and their abilities.

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The Tower of Hanoi involves many mathematical concepts, two of which are recursion and fractals.

Recursion: 

Recursion helps achieve the goal of moving all the disks from the first peg to the third in the least possible number of moves. In broader terms, recursion helps break down a large problem into smaller pieces for efficiency and accuracy. For the Tower of Hanoi, recursion is specifically used to continue making “towers” until the disk with the largest radius is able to directly move to the third peg for the other disks to stack on top. Therefore, the more disks, the more moves. By using the formula S = 2n − 1 (where S is the fewest number of moves to get all the disks from the first peg to the third peg and n is the number of disks), you can find the least possible number of moves for any number of disks n without physically solving the problem. 

Fractals: 

Fractals are smaller replicas of a larger idea, another concept that comes into play with the Tower of Hanoi. For example, in nature, fractals can show up like the petals in the leaf (replicating the leaf itself)! 

The Tower of Hanoi can be expressed as a fractal too, known as the Sierpiński Triangle.

The Sierpiński Triangle begins with an equilateral triangle, where all the angles of the triangle are 60 degrees, and all the side lengths are equal. Then, another equilateral triangle is formed in the center, after connecting the midpoints of the larger triangle (see the biggest white triangle.) Afterward, the triangle is removed, and three smaller equilateral triangles are formed where the same process continues again to infinity. 

How does this relate to the Tower of Hanoi?

This animation on the computer (link below) shows how the Tower of Hanoi is represented in the triangle: 

https://math.ucdavis.edu/~romik/downloads/hanoi-animation.gif 

The bolded red line shows the fewest number of moves with four disks. There are many other possible ways, not necessarily the fewest number of moves, to solve the puzzle through the triangle from the top to the bottom right corner, zigzagging through smaller triangles.

References

“Tower of Hanoi.” Encyclopædia Britannica, Encyclopædia Britannica, inc., 10 Nov. 2023, www.britannica.com/topic/Tower-of-Hanoi.  

Admin. “Solving the Tower of Hanoi.” Futurum, 16 June 2021, futurumcareers.com/solving-the-tower-of-hanoi.

Jost, Eugen. Beautiful Geometry. Princeton University Press, 2017.

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The Tower of Hanoi, also known as the Lucas Tower, was invented by the French mathematician, Èdouard Lucas, in 1883. Throughout his lifetime from birth on April 4, 1842 to death on October 3, 1891, Lucas was best known for his studies in number theory, particularly his work in the Fibonacci Sequence and his invention of the Tower of Hanoi.

The Tower of Hanoi is made up of:

Three long pegs

Eight to ten circular disks with different radii

The Two Rules:

Rule 1:  You can only put a smaller disk on top of a bigger one (not the other way around.)

Rule 2: You can only move the top disk in the stack from one peg to another peg.

The Goal:

The goal of the game is to get all of the disks from the first peg to the third peg in the least number of moves.

Start with 3 disks on peg 1 as a Tower of Hanoi, i.e., with the disks arranged in the order of smallest to largest radii (smallest being on the top of the pile).

Follow the two rules listed above and try to solve the puzzle, counting your moves.

After you have solved it once or twice, ask yourself, “can you do it again with fewer moves?”

Increase the difficulty by adding more disks when you think you have solved the puzzle with the fewest moves possible per n disks!

NOTE: PART 2 OF TOWER OF HANOI COMING SOON (with mathematical concepts involved)!! Try out the game online and see if you can spot any patterns/formulas!

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Summertime gives all of us a few months for some extra practice, training, and time to recharge!

Here are a few tips on how you can make the most out of your summer with math, while also resting and recovering!

Set a reasonable goal for yourself every day! You know yourself the best, so find time to have fun but also continue practicing. For example, many AMC fans would benefit from working on some of the past tests to prepare them for the AMC of their level during the school year. One strategy that helps me is to flag my incorrect answers and filter out the concepts that I need to work on for the rest of the week.

If you love summer camps, enroll in a few math ones! Not only will you get the opportunity to bond with math-loving people and create lifelong friendships, there will be a new chance everyday to learn new topics, strategies, and skills through mentors who have more experience and knowledge to offer. Some popular summer camps mainly for high schoolers are PROMYS, Awesome Math, SUMaC, and G2 Math Program.

Challenge yourself! Unlike the urge to get a perfect grade in school by avoiding risks, summer is perfect for pushing boundaries and finding out what you are capable of. For example, set time limits when attempting a set of problems, take advantage of the infinite number of math problems out in the world to try new ones everyday, and don’t be afraid to make mistakes.

Lastly, make time to recharge your brain after a long school year – it will sharpen your problem-solving skills and uplift your attitude! Hanging out with friends, listening to music, playing sports, and exploring new places are great ways to refresh and relax during the summer.

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Trust your preparation: You have already done all of the preparation for the competition; no need to stress or get anxious before the contest begins! You’ve got this!

Believe in yourself: If you are questioning whether you’re good enough, just know that you are more than good enough! It takes real courage and determination to register for a contest, let alone the time preparing for it. 

Pace yourself wisely: Don’t spend too much time on any one hard problem and solve the ones that are easier first. You will gain momentum and feel confident in yourself after knocking some of the problems out.

Read carefully: Read each question carefully. Skimming through questions is one of the most common ways to make silly mistakes and can be easily avoidable through patience and understanding the question word for word. 

Stay hydrated: Yes, really! It is proven that drinking water will increase brain function and critical-thinking skills. Also, getting a headache from dehydration during a competition will impact your problem solving abilities. 

Connect with mathematicians: Math competitions are an amazing way to make lifelong friendships, especially with people who share the same interests and love for math!

Enjoy the experience: During the contest, don’t think too much about the result or scores. As Dr. Alice Cortinovis once said, “If a math competition goes well, that’s great! If it doesn’t, who cares?” That being said, the key for success for any contest is to practice beforehand. 

Be curious: Approach the competition experience with an open mind. It is never too late to learn something new and staying curious will help you make the most out of the contest. Every competition is a chance to learn something new.

Set a personal goal: Setting a new goal for each math competition will boost your self-confidence, and you will truly use the competition experience to your advantage.

Have fun: Lastly, have a good time during the contest! Math competitions are supposed to be fun, so celebrate it while you have the chance.

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A few days ago, I got a valuable opportunity to be on my school’s Purple Comet Team! Purple Comet “is a free, annual, international, online, team, mathematics competition designed for middle and high school students that has been held every year since 2003” (https://purplecomet.org/). This year 3,752 teams from 75 countries registered for the competition, an incredible testament to the global love and importance for math.

Our team of six challenged ourselves in each problem with focus and determination and felt rewarded after solving them, sharing our ideas, and supporting one another. It wasn’t just about getting the right answers – it was about collaboration, critical-thinking, and bonding over our love of math. 

Personally, there was one aspect of the competition that could have been better; more girls on my school team and the competition. When I shared the list of team members with one of my math teachers, his first reaction was, “Oh! You’re the only girl.” It wasn’t because other girls weren’t capable of solving problems; however, the reason was more about the discouragement and self-doubt that prohibited them from competing. When I encouraged one of my math-loving friends to join, she frowned and said, “Mihika, I’m not good enough!” That was a sad moment for me as she is one of the smartest and talented people I know. 

One piece of advice I would give to all of the girls is that if I learned to believe in myself, everyone can do the same. Sometimes, I lack confidence and tell myself that I am not as knowledgeable or skilled, but I then remember that there are people who support me in everything I do, especially the math team. As I wrote in my earlier blog on Math + Girls = Fun, Dr. Cortinovis said, “If a math competition goes well, that’s great! If it doesn’t, who cares? So, participate because you have nothing to lose.”  

Also, team competitions are priceless ways to begin the math competition journey and gain experience because most members are uplifting, inspiring, and contributing to a positive team morale. So, take the first step. You are more than good enough.

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Sophie Germain was born on April 1, 1776 and later in her life, she became one of the most remarkable French mathematicians of all time. In her early life, Germain fought for her rights to education, enduring her parents’ opposition to learning and proving them wrong in the future. Even today, girls do not receive enough support and encouragement in mathematics and the majority of participants in math competitions are boys. From a Wall Street Journal article, published on January 5, 2025, Long the Star Pupils, Girls Are Losing Ground to Boys, the author wrote, “Boys now consistently outperform girls in math.” 

Germain helped narrow down the gender gap in her time by managing to find books and other resources about mathematics and other topics she was interested in. Over the course of her life, many people know her as the first woman to receive a prestigious award from the Paris Academy of Sciences’ on her work in the field of elasticity and for proving the Sophie Germain Theorem. However, the awards that Germain earned barely scratch the surface of all of her successes. 

She was also recognized for her work and inspiration from Carl Friedrich Gauss, a notable German mathematician, who worked together with number theory. Despite her talent, Germain had to disguise herself with the pseudonym of M. LeBlanc because a woman in mathematics was unacceptable by society at the time. Years later, she was the understudy of more famous mentors, such as Legendre, and developed her research in number theory and other mathematical topics. One of her quotes that inspired me was “It matters little who first arrives at an idea, rather what is significant is how far that idea can go.”

Sophie Germain is truly a female mathematician to be motivated by, from struggling to study mathematics even though it was discouraged to becoming one of the most noteworthy mathematicians. Her story taught me to always push for my rights to education and not allow anyone to make me feel inferior to them through my deep passion for math as her. 

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Have you ever wondered how astronauts were sent to the moon and back safely? Katherine Johnson, a female mathematician, was born on August 26, 1918 in West Virginia, and her work influenced NASA (National Aeronautics and Space Administration) and the discoveries of the universe. 

At a young age, Katherine’s love of math and number theory was apparent, despite her father being a lumberman and her mother as a school teacher. Katherine’s intelligence wound her up in high school at the age of ten! In West Virginia University, she was selected to be one of three African American students to take a graduate program in 1939. 

In 1953, she joined the National Advisory Committee for Aeronautics (NACA), now NASA, as one of the few African American mathematicians to solve problems, analyze data, and were a key part of the successful missions to the moon. Interestingly, the women were known as “computers”. Her calculations of placing spacecrafts, such as Freedom 7, Friendship 7, and Apollo 11, into orbit which helped open doors for NASA’s future missions and findings from space.

In 1986, Katherine retired and became a public figure, encouraging girls to work hard and never give up. Additionally, in 2015, she received the Presidential Medal of Freedom from the 44th president of the United States at the time, Barack Obama. “In her 33 years at NASA, Katherine was a pioneer who broke the barriers of race and gender, showing generations of young people that everyone can excel in math and science, and reach for the stars,” Obama said.

Katherine’s story inspires me to continue chasing my dreams, pushing through the gender gap and expanding the female math community.

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My school’s math team consisted of six students, two girls (including me) and four boys, to compete in the Caltech Math Meet (CMM). CMM is a math competition for high school students organized by the students of Caltech. We left school early to arrive at the San Francisco airport on time and headed to security first. It was the emptiest I have seen at an airport, so we had extra time on our hands to do homework and study for tests. After 1.5 hours, we boarded the plane, which flew us to LAX in forty minutes, the shortest flight of my life. From LAX, we drove one hour to the hotel, got dinner, and slept at 11:30 PM. At 5:30 in the morning, we woke up, showered, and walked to a breakfast place nearby. After packing our bags, we drove to Caltech and checked in. For the opening ceremony at 8:30 AM, we sat in the Ramo Auditorium for instructions on the individual and team rounds later that day. We headed over to Baxter for the ninety-minute individual round. The problems challenged me, but I felt accomplished afterward for solving many of them correctly! 

Outside of the Ramo Auditorium, the competition staff of Caltech provided lunch for all the teams. It was wonderful to see hundreds of other math lovers on a green lawn together, frantically discussing the problems from the first test. Additionally, I finally met the hardworking former president of Athemath, Vivian Loh, in person. As a member of the Athemath community, I would describe the organization as a brilliant way to draw together ambitious girls, the marginalized gender in mathematics.

As a team, we walked around the beautiful campus and headed to the next round, the proof-based team contest in Hameetman. There were many teams in the room, and I loved looking up to the way other members of my team brought unique insights to problem-solving. Even though we could not solve all of them in the allotted one hour, we tried our best to solve most of them.

After the team round (the last exam), we tried to reschedule our flight earlier, so we unfortunately did not stay for the awards ceremony. We also did not arrive at the airport in time; therefore, we decided to go to Sofi Stadium for team bonding and reward for the intense competition. After taking pictures of the stadium and Rivers Lake, we went to the airport for our flight. 

Overall, the experience was memorable, worthy, and rewarding. Spending time making new friendships and memories through a shared passion, gaining more math competition experience, and lots of laughs was what my two-day trip was all about.

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