CAMP 2022 Day 4

On the morning of Day 4, CAMPers had even more math puzzles to choose from – along with the regular linking puzzles, Hex, and Hanoi, we had miscellaneous math books – from The Moscow Puzzles to Eye Twisters – as well as the Magic Birthday Trick, scattered throughout the room. 

“It’s really simple – like, really simple. You just put one piece down and try to make a line between your two colors.”

Having found themselves locked out of their usual classroom, the CAMPers in SINE spent the first few minutes of Math class in the first-floor RKC lounge before switching to the Computer Lab (after the CS instructors had had time to “destroy the evidence” of what they would be doing in class today).

Once there, they continued to discuss fractal dimension and the Sierpinski TriangleWhy does it make sense for this object to have a dimension between 1 and 2? First, they reviewed the concept of scaling an object – for example, if you take a square of area 1, then scale each side length by 3, the scaling factor is 3. The ratio of the new area to the original area is 9:1 = 9/1 = 9 = 3^2. The exponent (in this case, 2) is the dimension of the object. Next, they took rulers and used them to draw the Koch Curve, before embarking on the quest for its dimension.

Meanwhile, the COS group continued working with fractional numbers (between 0 and 1) in different bases, specifically ternary (Base 3) – in which all numbers are written as strings of 0’s, 1’s, and 2’s. Which of these ternary numbers (e.g., 0.0121) is in the Cantor Set?

Once Math class was over, CSC headed to Art, while the CAMPers in SEC worked on problems in converting between bases (2, 3, 4, and 10) and adding numbers in Base 2 using truth tablesHow can we build a table with two inputs (A and B) and an output that is their Base 2 sum? What logical operators (AND, OR, XOR) give us these outputs? Given the materials they had been working with all week (cables, LEDs, and batteries), CAMPers were able to build their very own half-adder (which is able to add two 1’s together) and even made a truth table for a full-adder (which is capable of adding 1 + 1 + 1).

After lunch at Kline, the CAMP fragmented into groups for each elective: a hike to the Bard College Farm, lawn games and a thrill ride on the Circle Swing that hides in the shadow of the Campus Center, and a Rubik’s Cube demonstration by world champion speed-cuber, former CAMPer, former CAMP high school volunteer, Bard College math and physics major Daniel Rose-Levine.

At the end of electives, SEC went to Art class to make their own Sierpinski Pyramids.

Finally, the CAMPers got together in the Auditorium to puzzle their way through Rubik’s Cubes and to play a few more rounds of Rush Hour and Set.

Day 4 brought us tons of mathematical, computational, and artistic excitement. Now that we’ve had a glimpse of objects, dimensions, and number systems that seem less than normal (but are definitely real!), we’re finally ready for Friday, Day 5.

Photo Credit: Sonita Alizada (images 29-39, featured image), Kateri Doran (images 1-28, 40-42).

CAMP 2021 Day 5 – Time to Say Goodbye

We cannot believe it is CAMP Day 5 already! Welcome, all staff, parents, and CAMPers to Day 5. Today, before we head off to a new journey, we would like to introduce our CAMP 2021 Photo Gallery on the main page:

CAMP More Information

If you clicked into every small circle (Yes! Math Circles!), you will realize there are a larger photo AND small descriptions of each photo. These photos start from Day 1 to Day 5 and are kindly given to us by CAMPers, parents, and staff. 

We begin Day 5 with Karen‘s Computer Science class. Today, CAMPers are to build their own 3D model based off on the codes they write on Tinkercad.

CAMPers first went back to their codes to set off the background to be transparent to transform into a 3D model. Then, Karen introduces Tinkercad to CAMPers and they try it out themselves!

Although it is a new programming website, CAMPers are never afraid of trying out new ideas.

CAMPers start off from the original programming and changing the background into transparent like.

CAMPers are to make their borders clear for the model. The model can later be used for laser cut or 3D printing.


In Japheth‘s Math Class, CAMPers went through the history of tiles and tessellations. It begins with several mathematicians’ discoveries of the different tile sequences. Through different polygons, mathematicians discover several possible sequences. There are several types ranging with different combinations of polygons that built up to this research.

Over decades, mathematicians are finally able to come with a closer explanation of tessellations. While many remain found, there are still undiscovered patterns.

One famous mathematician, Martin Gardner, dives into the research of interesting graphics. This later becomes the turning point of tessellation discovery.

In Frances‘s Math Class, Sine and cosine students figured out that the sum of the degrees in the angles of a polygon can be found by multiplying 2 less than the number of angles by 180, (n – 2) x 180.

Although some had been told the formula, they were able to see why this formula works. All students worked to find more semi-regular tessellations, figuring out many of them and also making tessellations that they realized are not semi-regular, but quite pretty. Hope CAMPers can go back and share what these all are!

In Anish‘s Computer Science class, CAMPers design their own tessellation codes based out of Python Turtle (a continued work from yesterday!).  Today, CAMPers’ creativity is highly involved in this self-exploring, 10 minutes project. Anish gives all CAMPers 10 minutes to design a code that forms a tessellation. CAMPers are to create their own project using the knowledge they had before.

In Grace‘s Art Class, CAMPers create their own pull-up net. CAMPers slowly construct the foundation of the net, then forms a foundation of six cube sides. Then, they begin to color each side by drawing tessellations on each cube.

CAMPers create their own tessellation design on the cubes


CAMP Day 5 had been a short yet informative day. CAMPers enjoy their time with their beloved staff. While still having lots of time to use their creativity on every project.

Well done, 2021 CAMPers! Looking forward to meet you all next year!


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CAMP 2021 Day 4 – Tessellations Everywhere!

As usual, welcome to CAMP 2021 Day 4! As we are closing up the week, staffs and CAMPers remain energetic from the early morning. Today, CAMPers work on the in-depth application of tessellation. This includes: complex Voronoi Tessellations creation, constructing tessellations through mathematical understanding, and writing Python Turtle codes.

In Anish‘s Computer Science class, CAMPers learn how to build their own codes on Python Turtle. A Turtle is a pre-installed Python library that allows CAMPers to create pictures and shapes through a virtual background.

Anish’s class begins with a Python Turtle’s introduction.

CAMPers construct their own Turtle codes, take close looks at the Turtle, and write down their guess on what the Turtle will turn out to be.

CAMPers guess what the Turtle codes will end up like.

In addition, CAMPers begin their own journey on Turtle. They write down codes that will lead the Turtle to draw a rectangle in fixed length and width. The turtle runs from one edge to another, CAMPers are to set certain length for it to run.

CAMPers learn how to write a Turtle code to draw rectangles in fixed measurement.

CAMPer’s work on Turtle

In Frances‘s Math Class, CAMPers work on Tessellation Creator, a website that helps CAMPers to build their tessellations online. CAMPers are given information is yesterday’s class about polygons. It is now the time to fit every shapes into one!

CAMPers create Tessellations based on polygons

Student work – create tessellation digitally

In Karen‘s Computer Science Class, CAMPers are introduced to the vertex in programming. They learn how the code endshape” is used to end every shape they create. With this code, many shapes can be created through one canva. Notice how the code “funshape” repeats itself over and over time, the different rotation of the shape itself helps each one fits into one another (just like a puzzle!). 

CAMPers explore the codes and figure out the vertex of each shape

Later on, CAMPers draw shapes and fill out their colors. They create shapes that corresponds with each other – together becomes tessellations

Students can create multiple shapes in the canva

In Grace‘s Art Class,  CAMPers learn about the literal meaning of polygons. They also learn about Voronoi Tessellation – a random sequence of patterns. For instance, city and map views. That said, the World map is also a Voronoi tessellation (see image). 

Definition of polygons!

Our map, if found the right spot, is also a Voronoi Tessellation!

What are some Voronoi Tessellations in real life do you have in mind?

CAMPers working on their Voronoi Tessellations!



Last but not least, we wrap Day 4 with — Among Us! In Ayanna’s Activities, CAMPers enjoy a Mathematics Among Us that is looking for imposters.

As we are getting closer and closer to our last day, CAMPers are more familiar with tessellation and its applications! In the end of the week, we are not ready to say goodbye to CAMP, but what we learn will always carry along with us.


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CAMP 2021 Day 3 – Half Way Through

Welcome to CAMP Day 3! CAMPers had worked their way through the week with hard work and creativity. Today’s work focus on the application of tessellations. Each class explores topics on tessellations with CAMPers experimenting with each subject with their hands.


In Shuang‘s Art Class, CAMPers begins with exploring different real-life tessellations. For instance, bee comb, turtle’s shell, pineapple, and tiles.

CAMPers working on real-life examples with tessellations


CAMPers later have a conversation on “Why does tessellation exists?” They give out several reasons: natural efficiency, saving space, and having an arch-like shape like a turtle shell to provide protection

CAMPers are also introduced to Voronoi Tessellation. Unlike the tessellations CAMPers encountered before, Voronoi Tessellation spreads out in a less equal pattern yet still having connections on a plane. For instance, a giraffe’s skin will be a Voronoi Tessellation.

To make the objective clear, CAMPers play a supermarket imagination activity that Shuang designed. This activity involves CAMPers imagining themselves in a supermarket and making various points outward.


Voronoi Tessellation is different from the tessellation CAMPers discussed


CAMPers later designed their own chicken pattern through origami papers. Chicken patterns design is also used in this year’s CAMP T-shirt (by Shuang!).

CAMPers made their own chicken shape design

CAMPer working on the self-created tessellation book

In Grace‘s Art Class, CAMPers also work on chicken patterns making. CAMPers also learn how to fold chicken pattern from scratch and to construct several chicken patterns together to form a tessellation.

Chicken Pattern Tessellation


CAMPers also learn how to make their own tessellations through reconstructing and connecting each pieces together.

CAMPers learn how to make their own tessellation


In Karen‘s Computer Science Class, CAMPers write codes to form images. They then translate the images they create and create the shape over and over again. Eventually, they make multiple shapes that form tessellations. 


CAMPers write codes on translating the image they create

By Rotating shapes, CAMPers create different patterns

The final product – tessellation!

CAMPer’s tessellation work

CAMPer in Karen’s class working with tessellation codes


In Frances‘s Math Class, CAMPers in SINE and COSINE are figuring out how to find the sum of the angles of a polygon using the fact that each triangle’s angles sum to 180 degrees.

CAMPers working on degrees in polygons

CAMPers learn how to divide polygons into triangles

Unlike going straight to definition, CAMPers have their hands on experience on testing out different angles to see how different polygons work. After these observations, CAMPers calculate what each angles are and label out each angles of the polygons.

In addition, CAMPers in secant and cosecant have figured out
       1) The number of degrees in each angle of a regular polygon
       2) That each vertex of a tessellation must be surrounded by polygons whose angles add to 360 degrees
Next they began to make use of the rules to create tessellations with more than one shape of polygon.
In Japheth‘s Math Class, CAMPers work on solving polygon tessellation around a point problem. CAMPers are introduce to both geometry and algebraic ways to understand tessellations.

Ways of understanding the polygon tessellation around a point problem

As we get to the middle of CAMP, CAMPers continue to show effort and passion in the topic being introduced. Through activities and experiment, math becomes a fun activity and knowledge for all!


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CAMP Day 2 – The Process

It is CAMP Day 2! After having a bright start yesterday, Day 2 begins with CAMPers receiving more knowledge on tessellation.

CAMPers received the CAMP 2021 T-shirt

In Frances‘s Math class, CAMPers created tessellations using a scalene triangle and then examined them to see how they demonstrated that the sum of the angles of a triangle is 180 degrees.  CAMPers then went on to use that information to figure out the sum of angles in any polygon.

Student’s work on tessellation


Student’s work on tessellation based on angles and triangles


CAMPers are asked for what pairs they see on their tessellation works: “Where do the six triangles meet?”

They also label out their works and present them in class. Frances introduces to CAMPers the definition of polygons and different types of shapes (ex. hexagon, octagon, twenty-two gon). 


CS class begins with a review on GeoGebra


In Computer Science class, CS Instructor Karen introduces p5js, a website that allows students to use Javascript to run codes. It allows students to write their own code and run it on the screen


p5js allow CAMPers to run their own codes


CAMPers are introduced to coding languages such as:

function setup ( ){createCanvas(___, ___);}

CAMPers are also introduced and tried out RGB color scheme. RGB stands for red, green, and blue. RGB decides what color it is and gives unique codes to each color. For instance, certain color green will have certain codes. 


RGB color scheme allows different colors to be displayed on the screen


CAMPers also dive into the Pantone website for different color schemes. They later go back to the background color code on p5js.  Background (R, G, B) with RGB set up will give different colors on the screen.

In Shuang‘s Art Class, CAMPers start designing works and prepare for their work presentation. CAMPers create their own collage by ripping paper polygons but not the corner. 


Polygons that will result in different forms, remember not to rip the corner!

CAMPers designing their own collage through the pieces they made before


With the principle, CAMPers can try out different polygons to see what the polygons bring to them. It might bring out different results! A right polygon might bring out a different result with a scalene polygon.

In Grace‘s Art Class, CAMPers talk about how tessellation patterns are used throughout history. CAMPers are introduced to Leo Ming Pei, an artist, and architect who designed the glass pyramid entrance of the Louvre (tessellation!).


Triangles guess game CAMPers enjoy in Grace’s art class


CAMPers browse through images of Mesopotamia, Paris Louvre, Moroccan tiles, and Brazil sidewalks as their art inspirations. In hands-on activities, CAMPers work on their own tiny Louvre tessellation for Zine. The tessellation is built on cubes by use of rotation.

We end Day 2 with many activities that revolve around tessellations, CAMPers are now more familiar with the real-life examples of tessellations!

CAMPer working on an art project


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CAMP 2021 Day 1 – Journey Begins

Today is Day 1 of CAMP! CAMPers join us all around the region this morning. After last year’s virtual CAMP, we decided to host this year’s CAMP online too. Our theme this year is Tessellation.  Tessellation is when a surface is covered by shapes that have no overlaps or gaps.

CAMPers are divided into 6 groups: SIN, SEC, TAN, COS, CSC, and COT, the six functions of angle commonly used in trigonometry. The groups are divided by their grades and the math packets they did during application. The first session (SIN, SEC, TAN) begins from 9:00 am – 1:45 pm, while the second session (COS, CSC, COT) begins from 11:00 am – 3:45 pm. Classes include Computer Science, Math, Art, and Activities.

Beginning from Computer Science Class in the morning, CAMP CS Instructor Karen Blumberg introduces an online tool name GeoGebra. GeoGebra is an interactive geometry, algebra, and statistic application that allows CAMPers to design their own tessellation models.

Karen later introduces the rule that with any segment AB and a randomly located point C, there will always be a connection between segment AB and point C. CAMPers test the proposed rule itself on GeoGebra. 

Segment AB and point C on GeoGebra.

Later on, CAMPers work on forming parallel lines with segment AB that passes through point C on their own screen. The goal for this model is to eventually form a parallelogram. 

CAMPers forming parallel lines with segment AB, connecting point A and point C together to form segment AC.

Final Goal: To create a parallelogram using GeoGebra and the listed rules.

In the afternoon Computer Science Class, CS Instructor Anish Anne leads CAMPers to use GeoGebra to form a tessellation based on parallelograms.

Anish guiding CAMPers to form tessellations


CAMPer’s work in Anish’s class


In Art Class, Art Instructor Shuang Cai introduces how to use origami papers to make an isosceles triangle. CAMPers are given a mission to fold an equilateral triangle using square paper.

Shuang also introduces the final product of CAMP’s Art course – students will create their own Tessellation Zines to share. A zine is a small self-publish work that is full of images or text. In today’s class, CAMPers start making the skeleton of their zine. They learn how to fold an 8-page zine and are shown with an introductory video by Shuang. 

CAMPers learn how to fold their own 8-page zine


Click to watch: Shuang Video

In Math Class, Math Instructor Frances Stern introduces the definition of tessellation to CAMPers. CAMPers are introduced to several examples of tessellation and begin to complete a tessellation of a scalene triangle.

In Activities, HS Volunteer Shoshi Cohen leads a set game with fellow CAMPers. CAMPers are to observe the given patterns and select the three that share similar rules. The rules could be sharing a similar shape or color.

CAMPer enjoying the set game

CAMPers found the sets before through observing the rules


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CAMP 2021 Staff Page

Welcome to CAMP 2021! Here is our amazing staff.

Japheth Wood
CAMP Director and Math Instructor

I’m Japheth Wood (he/him), the CAMP director, and a math professor at Bard College. I can't believe that we’ve reached our eighth summer of CAMP! We've been recognized a second time by the American Mathematical Society with an Epsilon Award. What a nice honor, on top of our original support from the Dolciani Math Enrichment Program. We are online for a second summer. While it is heartening to see the resiliency of our students and staff to meet online, I am very much looking forward to the time when we can all safely return to the classroom and learn together in person, with online interactions for convenience.

Frances Stern
Co-Director and Math Instructor

My name is Frances Stern (she/her/hers)! I’ve been teaching math at CAMP since the first year, sometimes choosing the program’s math theme. It’s fun to show students math that they don’t see in school. During the school year, I teach math part-time at a private school in New York City. My hobbies include learning to paint & juggle, reading, walking & folk dancing. I’ve written 2 books for teachers and parents called “Adding Math, Subtracting Tension” (for 2 different age ranges of children). They pay as much attention to how to keep out of a fight (what many parents told me is a problem) as they do to math.

Talia Willcott
CAMP Coordinator

Hello! I am Talia, the CAMP Coordinator. I am here to help with any questions or concerns that may come up. I am currently studying math and physics at Bard College and am going to be a senior this upcoming year. I have worked as a tutor with the Bard Math Circle for about two years now and am always looking for new opportunities to connect with people who are interested in math! My hobbies include playing music, making art and solving problems. I love working with students and am dedicated to helping them find their passion! 

Grace Villamil
Senior Staff / Art Instructor

Hello My name Grace (she/her/hers).  I am a multidisciplinary artist, blending different mediums (such as sound, material, drawing and video) to create altogether new forms. I love adventuring into new realms, and discovering inventive ways to express a thought or an idea. I am Filipino-American, I am originally from California ~  lived in NYC for over 15 years and now live in Bearsville, NY.  I love swimming, outer space/the universe and discovering what answers can be found in nature, (so SO many!) This is my first year with the Bard Math Circle Summer CAMP. I am so very excited to meet you &  be your Art Instructor this summer ! Please check out my website for my interstellar installations I created in such places as NYC and Berlin:

Karen Blumberg
Senior Staff / CS Instructor

I’m Karen Blumberg, and I’m excited to be a CS teacher at camp this summer! During the year, I’m a Math Teacher and Technology & Innovation Coordinator at The Brearley School in New York City. I work with faculty and students to integrate technology academically, creatively, and responsibly into their classwork, and I help support projects that include programming, web design, robotics, 3D design and printing, digital art, and more. I studied math in college, and I’ve always loved math puzzles. I encourage my students to think of themselves as forensic mathematicians in order to consider all the clues in order to solve a problem efficiently. Over the years, I’ve been introduced to a few different computer languages (Basic, Pascal, C++, Lingo, Logo, Processing, Java Script, Python, etc.) Besides teaching and learning, I love traveling, tasting new foods, and taking photos.

Shuang (Chelsea) Cai
Senior Staff / Art Instructor

My name is Shuang Cai. This is my third year working with Bard Math Circle CAMP, and this year I will be joining as a senior staff on this team. I am currently a rising senior at Bard studying computer science and studio art. I am coding on my own game with Unity. Maybe you can try it out when I’m done?

Anish Anne
CS Instructor

Hi, I’m Anish (he/him/his). I am a rising sophomore in Upstate New York. I am very excited to be part of the CAMP staff this year and have been involved with the Bard Math Circle since taking the AMC 8 exams in 6th grade. Since then I have participated in several other exams as well as been a camper at CAMP. My hobbies include playing tennis, chess, and programming. This year I will be teaching Computer Science at CAMP.

Annie Moulene
Junior Staff / TA / Zoom Hosts

Hi! My name is Annie Moulene (she/her/hers). I am a junior studying Chemistry at Bard. In high school, I loved doing math. I was always one of the students that other students would come to for help since I could help break down the question and show them the steps needed to solve it. I love trying to connect problems in a way that it becomes applicable to the real world since that helped me understand and appreciate math more. I’m excited to be organizing our zoom room and I look forward to helping you solve those challenging problems and helping inspire a new love for math.

Ayanna Battle
Junior Staff / TA / Zoom Hosts

My name is Ayanna Battle (she/her), and I’m a rising senior studying music composition and math at Bard College. This is my first year as a junior staff member at CAMP, so I’m really excited to meet everyone and learn more about CAMP and CAMP culture! For the past couple of summers, I’ve worked at a summer enrichment camp in Atlanta, GA and taught math to rising 8th graders, so I’m looking forward to seeing the similarities and differences between the two programs!

Hadley Parum
Junior Staff / TA / Zoom Hosts

Hello! I’m Hadley Parum (they/them/theirs), and I am a graduate from Bard College (’21) in Psychology and Music. I am excited for my first year as a junior staff member at CAMP, and hope to share an interest in approaching a diverse range of problems with the tools of mathematics and a zeal for questions. I’ve worked extensively as a tutor in areas of math, coding, writing, and music theory, and have found exciting connections between communication,  mathematics, and art. In addition to contributing my own passion towards topics in these areas, I look forward to hearing what most excites others during the program!

Pin-Shan Lai
CAMP Social Media Coordinator

I am Pin-Shan Lai (she/her), the social media coordinator. I am a rising sophomore majoring in political studies and literature at Bard. This is my first time joining CAMP and I am very excited to meet everyone in the upcoming weeks. As a non-math major, I enjoy working with mathematics and science outside of class. I spend my time working on food science and learning about chemicals.  I enjoy cooking, planting vegetables, and reading both fiction and nonfiction novels.

Santanu Antu
Junior Staff / TA / Zoom Hosts

Hello! I am Antu (he/him/his). I am a rising Junior at Bard College, majoring in both Math and Physics. For the last few years, I have invested a lot of time exploring various branches of mathematics and their applications in theoretical physics. My endeavor in this labyrinth of equations and theorems helped me recognize a serious flaw of modern mathematics and physics- the lack of communication. This unpleasant issue is not only obstructing the development of these captivating fields but also leading to mass scientific illiteracy. My goal for the Math CAMP is to communicate some of the complex ideas of math in a nice and simply accessible way. I’m looking forward to organizing the zoom meetings, think about math puzzles, and spread the message- “Mathematics can be fun.”

Shoshi Cohen
Junior Staff / TA / Zoom Hosts

HI! My name is Shoshi (she/her) and I am a senior at Avenues: The World School. I have been participating in Bard Math activities for quite a few years now, including Bard Math Circle and Bard Math CAMP. This year I decided to revisit the fun I had in attending these events by working as a counsellor for Bard Math CAMP! I am so excited for this year's CAMP and I hope it's a blast! 

Tina Giorgadze
Junior Staff / TA / Zoom Hosts
Hi! My name is Tina and I am a rising Junior at Bard college majoring in mathematics and computer science. This is my first year at CAMP, and I am beyond excited to work with students eager to learn mathematics and its applications in art. I have been a Bard Math Circle tutor for two years now. I always have a wonderful time brainstorming new ways to communicate math solutions and concepts. I  look very forward to hosting fun and engaging activities over our Zoom sessions! I am happy to have the opportunity to use my experience to help CAMP participants tackle math puzzles and interesting problems.

CAMP 2020 Day 5 – again next summer

Today is our last day at CAMP 2020. We felt heartbroken to say goodbye to all the CAMPers. It was wonderful having you with us this week: the time we spent together was truly a highlight of our summer, notwithstanding we were all staying home.  

CAMP team at our last staff meeting

  • Math with Japheth, Erin, and Frances: At 1 PM in Japheth’s class, the COT group approached the second to last class for the entire program. After their discussion on the Hill cipher and two by two matrices modulo 26 on Wednesday and Thursday, the group focused on analyzing invertible ciphers and matrix ciphers.

If y = f(x) = y and we have g (f(x)) = g (y) = x, we call g (y) the inverse function of f (x). Not all ciphers were created equal, some like the MULT-2 (Multiplication cipher times 2) does not have an inverse function corresponding to it. CAMPers discussed the possible invertible ciphers from the various kinds they had become familiar with over the week. Later we went into the breakout room for more mathematical problems and group discussions. 

CAMPers were also curious about what websites or apps Japheth uses to type math equations and symbols efficiently. Our instructor shared a few options (Overleaf!) he knows to encourage students to explore math further after CAMP ended.

A few quotes from the CAMPers describing the program

Our senior math instructor Erin, teaching the COS and SIN sections, rediscovered a code that she shared with a 7th grade class years ago. Her class decrypted the first few parts of a text created with the intention to be easily decoded… by aliens! It’s an Active SETI (Active Search for Extraterrestrial Intelligence) message that was sent as an interstellar radio message in 2003 (directed toward nearby star systems). Scientists were hoping that extra-terrestrials could potentially decode the message and learn a bit about earth—what a fun background behind the history of this code. 

  • Brute-Force algorithm in programming: At the last classroom period with CAMP 2020, CS instructor Karen helped students perfect an encoding/decoding game. For the last few days, CAMPers studied various functions in the Python language, coding a program to encrypt and decrypt messages quickly with the given keys.

CAMPer Arvind was interacting with his classmates

Today, we studied the Brute-Force algorithm — one way to decode the message without the key. Although this method has a clear bias for English speakers with spelling ability (we still need to hand-select the original plaintext from 26 choices of keys), the Brute-Force algorithm is commonly used for examining the security-level of crypto-systems

After watching Karen’s demo on screen, students all coded their own cipher game ready to be tested. CAMPer Advika offers the group to run a secret message she made yesterday. Among all the possible plain texts, we noticed that “hovercam” seems to be the correct answer. 

  • Knot cipher: Last day in the art classroom, we designed a knot cipher with art instructor Tiffany and Chelsea.

Can you decode the message?

  • During the activity period, our team sent out a survey for their experience at CAMP before students played games with their TAs. We inquired about whether they were comfortable with the level of material, the pacing of the class, and what they liked best about CAMP to help us improve in the next summer. 

It was a lovely but short journey with CAMPers and families this year. Hope to see you again next summer. 

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CAMP 2020 Day 4 – “we don’t want CAMP to end”

We almost didn’t want to announce that it is Day 4 at CAMP 2020. Tomorrow will be the last day CAMPers spend time with new friends in the Zoom classrooms, eagerly sharing ideas and stories. Over the past four days, students have bonded with their classmates, their mentors (our junior staff members) assigned to each group, and senior instructors.

  • 10 AM EST, Art class with the SEC group: [August 2021 – This paragraph was updated to offer a more accurate description of Tut.] Tiffany, our art instructor, started the course with a fun activity. Tutnese, Tut, or Double Dutch, is a language game invented in the U.S. by enslaved Black People. Historically, Tut was used first as a way to communicate with each other without detection by slaveowners. Later, Black children also learned Tut in order to converse in privacy from non-speakers, including adults. For children in the twentieth century, it became a game. In her autobiography, I Know Why the Caged Bird Sings, author Maya Angelou (1928–2014) recalled trying to learn it instead of Pig Latin during her childhood. Tutnese was essentially a spoken crypto-system that was publicized by African-American Gloria McIlwain in her 1995 book which explains the language. It has experienced a renaissance in recent years through the internet. In Tutnese, vowels are pronounced normally, but each consonant is replaced with a syllable from the following table Tiffany shared on Zoom:

We went around the Zoom classroom, and CAMPers all pronounced their names in Tutnese.  

After the refreshing starter, the SEC group proceeded to make artwork with secret messages. CAMPers encoded a fun fact about themself with the abstract word alphabet they made on Day 3. In other words, we substituted a symbol for every letter. CAMPers used various mediums of their choices like watercolor and crayon. With grid-like composition and colors, their message assembles modernist paintings and patchworks. If we weren’t provided with the keys, it would be nearly impossible to decrypt the text from visual images.

Tiffany was showing her demo on Zoom


A message from CAMPer Sarah G, the TAN section

The key to Sarah’s text

After collecting works from students, Tiffany shared a story at her house after the storm on Tuesday. Her rain gutters were somehow filled with cute tadpoles

Tiffany shared a story happened at her house

Captivated by the story, A CAMPer led the conversation on how long tadpoles develop into frogs. He was so excited to hear from our instructors again soon: sending another photo to update the event that he forgot that our program ends tomorrow

  • 11 AM EST, Math with Erin Toliver: Soon after meeting Erin weeks before the program, I learned that she is a Bard alumna. This year she is returning to Bard on her 20th anniversary after receiving her master’s in math at Dartmouth College. Just today, we heard that CAMPer Gemma from the SIN group adores Erin a tremendous amount.

I asked Erin for her perspective on the program and why she devoted so much time and effort to teaching math with children. 

“I’m so grateful for the opportunity to be involved with the Bard Math Circle! Though it wasn’t around during my student days at Bard, an amazing community has developed since. I especially love sharing my passion for mathematics with kids because they’re so open to new ways of looking at things. I love seeing the look on a student’s face when they’ve discovered a new pattern, found a different perspective, or made a new connection for a deeper understanding of this glorious world of mathematics.”

— Erin Toliver, CAMP math senior instructor

Erin, wearing the CAMP t-shirt from 2015

Erin’s dedication to mathematical education for youth was undoubtedly reflected in today’s class. After meeting the CAMpers on Day 1. Erin quickly adjusted her plan for bettering students’ learning experience with CAMP.

In her classroom, we analyzed a long encrypted message through letter frequencies and common letter pairing. With the help of Tsitsi Mambo, an undergraduate at Bard and the TA for the COS section, Erin made sure she engages all of the CAMPers from multiple devices, including her iPad, iPhone, and computer. 

CAMPers were decoding a few fun codes on the printout through Erin’s iPad.

Nora from the COS group was the first student to bring ideas on the secret message. She observed from letter frequencies that the “MIT” in the text would be “the,” and “MIAM” should stand for “that.” We also heard from CAMPers Ben, Joshua, Vi, Annika, Jason, Mia, and Zozo, who hesitated to open the camera since they had lost power in the storm, and were using public WIFI today.

By the end of the class, we accomplished the goal of decoding the message: “I think that I am among the few lucky ones who are exploiting complexity. Most people are unhappy with the emergence of complexity. They would prefer it if the world were very simple, but then it would be a doom for a cryptographer like myself”Adi Shamir, a co-inventor of the RSA algorithm. 

  • 2 PM EST, Programming with the CSC section: I joined the CSC group in the afternoon with CS instructor Karen Blumberg. Before the class, Karen asked CAMPers what they had been up to at the math and art classes. Rafa and Gaia responded to Karen saying they were learning Modular Arithmetic and encoding with art. 

The group further developed their Caesar cipher algorithms into an encryption/decryption game that we can play multiple times with different plaintext and keys. Our instructor shared some shortcuts of coding on the screen. 

CAMPers were coding a cipher/decipher game

As usual, Arvind and Elias shared their experience with the Python language on their screen. A few of the less talkative CAMPers from Day 1 also became more verbalized, raising their hand for suggestions to improve the design. Each student from the group seemed confident about the new skill they learned, together they made a spectacular collection of minds (a quote from Karen). 

Alexander Warren, CAMP alumni (he was a student in the very first year of CAMP!) and a rising sophomore at MIT, are now working as the junior staff for the COT group. He majors in Computer Science and Electrical Engineering at school. In the past few days, Alex befriended all members in his section: he brought up some issues from working with multiple programming languages. Although a negative value works in the Python language, it would create an error in C#. We also knew that there is a thousand times limit with the Python: the game CAMPers designed can only run a thousand times before shutting down. 

  • Though torn between sadness and excitement for our last day, we also had some good news at our staff meeting: CAMP was recognized by the American Math Society with a 2020 Epsilon Award (link to This award included financial support that allowed the “Creative and Analytical Math Program” of the Bard Math Circle to provide need-based scholarships to CAMP this summer. We hope that more and more CAMPers can join us next year, and in years to come.

Hope to see you once more tomorrow. 

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CAMP 2020 Day 3 – after the storm

Time passes remarkably quickly when each day is fulfilling. It is already Day 3 at CAMP. After some anxiety over the storm yesterday, our fantastic team felt recharged and ready for some more math, CS, and our CAMPers’ inimitable creativity.

  • Uniting Communities: Long before our daily start time of 9:00 AM, our coordinator, Felicia Flores, started receiving emails from parents. Many CAMP families were still experiencing power and internet outages due to Hurricane Isaias.   

Shaylen from Hopewell, NY, told us that he had to work from the passenger seat of his family’s car and from Panera, sitting outside in the heat. Nonetheless, he missed his classmates from the COT group. Several more students, such as Eamon from the SIN group, worked from their neighbors’ backyards or houses. We were impressed with CAMPer’s commitment to continue learning and felt rewarded for our decision to keep the program running online this summer. 

Shaylen played SET with his classmates in the car.

 Even though we managed to keep all the classes running yesterday, some students were unable to join us. They asked us for notes of the courses they missed and were eager to catch up on work. Starting on Day 2, our instructors began putting in extra hours to summarize and upload notes for classes to CAMP’s shared Google Drive. 

Advika from the CSC group during class

  • Live Interactive: With the CSC group at 11:00 AM, senior instructor Frances Stern hosted her math class. Today, we discussed the period of Caesar cipher keys and the multiplication cipher with key 3. Recall that when we were studying the Caesar cipher with a shift of 13 (ROT-13), we realized that if we applied the change twice, we returned to the original plaintext. A “period” of the function thus means the numbers of repetitions we need to return to the original message. What if we try to decide the period of each of the possible Caesar cipher keys? 

When I dropped-in for Frances’ class, her students were enthusiastically raising hands. CAMPers proposed their answers for each of the Caesar cipher keys, explaining how they got their answers. Soon, the group moved to the multiplicative cipher with key 3. CAMPers encrypted the individual letters to cipher text. When Raphaela was asked how she got the answer for the letter s (18), her classmate Isa chimed in and helped her explain. 

CAMPers participating in math class on Zoom

A highly engaging classroom environment is inseparable from our instructors’ teaching philosophy. Through communicating with parents on a daily basis, CAMP wants to ensure that students keep their cameras open during class time. To increase the attention span of students, we also fit in a break period between each class. 

Frances, who has published two books on doing math with kids (please link this to:, wrote the following on her teaching philosophy:

“All children can learn and enjoy mathematics if they are engaged at a level appropriate to their understanding and knowledge with the purpose of helping them to build upon these. They like to know why they are learning a topic, so starting with an interesting problem – something they do not know how to solve – and allowing them to find solutions engages them creatively while demonstrating the “why.” That can be followed with practice – questions – to gain fluency and efficiency.”

A photo from the day Frances earned her juggling badge

  • Advanced Math in Programing: Next stop is the CS classroom, students were facing some new challenges. Remember that we made a Caesar cipher with the Python language yesterday? Today, CAMP upped our game with encoding matrices as lists, performing matrix multiplication, and operating on column vectors. Since most CAMPers have not learned about matrices at school, instructor Matt Goehrig opened his class by introducing students to the mathematical concept. 
  • Art and Cryptography: In the art classroom today, instructors Tiffany and Chelsea let students explore abstract art cipher in physical, or digital settings with Pixel studio. CAMPer made a pictorial alphabet using graph paper or the app. Using their unique visual alphabet, CAMPers will then create an artwork containing a secret message. 

“DONT EAT THE CAT” from our CAMPer Ben

A secret message from Arvind. Did you figure it out?

See you again soon. 

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