2021 AMC 10/12B Results

Here are the overall results for students who took the 2021 AMC 10B and AMC 12B contests at Bard College (online):

School AMC 12 Statistics

  • Average score for entire school is: 84.9
  • Average score for grade 12 is: 74.3 (4 Students)
  • Average score for grade 11 is: 79.5 (4 Students)
  • Average score for grade 9 is: 117.0 (2 Students)
  • AMC 12 School Team Score is: 393.00 (Team is defined as the 3 highest scores)

Top AMC 12B Scores at Bard

  • Shreyas Kar, 136.5 points (11th grader, age 16, duPont Manual High School)
  • Adrian Richard C Salazar, 130.5 points (12th grader, age 18, Philippine Science High School – Bicol Region Campus)
  • Anonymous, 126.0 points (9th grader, age 14)
  • Dong He, 112.5 points (11th grader, age 17, Homeschool)
  • Shraman Kar, 108.0 points (9th grader, age 14, duPont Manual High School)
  • Anonymous, 96.0 points (12th grader, age 17)

School AMC 10 Statistics

  • Average score for entire school is: 65.1
  • Average score for grade 10 is: 71.2 (9 Students)
  • Average score for grade 9 is: 68.3 (11 Students)
  • Average score for grade 8 is: 66.3 (6 Students)
  • Average score for grade 7 is: 38.0 (3 Students)
  • Average score for grade 6 is: 48.0 (1 Students)
  • AMC 10 School Team Score is: 333.00 (Team is defined as the 3 highest scores)

Top AMC 10B Scores at Bard

  • Andrew Carratu, 124.5 points (8th grader, age 13, Areteem Institute)
  • Anonymous, 109.5 points (10th grader, age 15)
  • Anonymous, 99.0 points (9th grader, age 15)
  • Anonymous, 99.0 points (10th grader, age 15)

Solutions to the 2021 AMC 10 and 12 contests are available on the Art of Problem Solving webpage:

Schools Represented

  • Areteem Institute
  • Bard Academy at Simon’s Rock
  • Cambrian Academy
  • Christian Brothers Academy
  • Crompond Elementary School
  • duPont Manual High School
  • East Eidge High School
  • Edgemont Junior-Senior High School
  • Gifted Math Program at University at Buffalo
  • Homeschool
  • Kingston High School
  • New Paltz High School
  • Philippine Science High School – Bicol Region Campus
  • Princeton Unified Middle School (PUMS)
  • Red Hook High School
  • Riverdale Country School
  • Spackenkill High School
  • Todd Middle School
  • Valley Stream North High School
  • Van Wyck Junior High School
  • Wappingers Junior High School
  • Williamsville North High School

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. 

For more updates: follow us on Facebook (https://www.facebook.com/BardMathCircle/), Instagram (https://www.instagram.com/bardmathcircle/), and Twitter (https://twitter.com/bardmathcircle)!

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: Tiffany, our art instructor, started the course with a fun activity. Tutnese, or Double Dutch, is a language game invented in the U.S. Historically used by children, who apply it to converse in privacy from adults, it is essentially a spoken crypto-system. 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 https://www.ams.org/prizes-awards/paview.cgi?parent_id=3). 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. 

For more updates: follow us on Facebook (https://www.facebook.com/BardMathCircle/), Instagram (https://www.instagram.com/bardmathcircle/), and Twitter (https://twitter.com/bardmathcircle)!

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: https://www.talkaboutmath.org/), 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. 

For more updates: follow us on Facebook (https://www.facebook.com/BardMathCircle/), Instagram (https://www.instagram.com/bardmathcircle/), and Twitter (https://twitter.com/bardmathcircle)! 

CAMP 2020 Day 2

Welcome to Day 2 at CAMP 2020. It is raining in most of the places where our CAMPers are. What a chilly day to stay dry, make new friends, and learn cryptography under creative settings! 

Things started coming together today. Our instructors prepared different plans for their groups. In the TAN and COT sections of Computer Science, Matt Goehrig reviewed some commands with his students. They discussed Numify and the If statement before revealing the topic of Day 2: creating your own Caesar Cipher

Students in Zoom Classrooms

Caesar Cipher, or Caesar’s Cipher, refers to one of the most straightforward and widely known encryption techniques. It is a type of substitution cipher. Each letter in the plaintext is replaced by a letter with specific predetermined numbers of positions down the alphabet.

Although students have learned how to manually encrypt and decrypt with Caesar Cipher, programming the cipher into digital code still provides a lot of conveniences. If CAMPer wants to quickly encode a long plaintext to their friend, they can now build a computer program. This also enables the recipient to decode a complicated cipher text without hours of labor. 

To program a Caesar cipher into the Python computer language, Matt asked the students to build a flow chart of their plan, adding the functions they needed and putting each command in the right sequence.

  1. Turn the message into numbers
  2. Add the secret number to each number 
  3. Turn that into letters 

At the COT group, Matt chose the message “BardMathCamp” and the secret number 11 as examples during his demo. CAMPers utilized the commands they learned yesterday to successfully synthesize them into a pair of encryption/decryption algorithms: a simple Caesar Cipher program. The group also modified the code to translate complete paragraphs of plaintext.

Senior Instructor Matt Goehrig teaching how to make a Caesar Cipher on Google Colaboratory

For the art class, SIN, CSC, and COS sections created Zoom backgrounds with Pixel Studio, a pixel art editor commonly used by artists and game developers.

Student works from the CSC group (Top to bottom, left to right: Nora, Instructor Chelsea, Rephaela, Elias, Arvind, Advika, and Isabella)

We asked the students to install the app before our class. So that they can get to know the app and be prepared for future courses, Chelsea Cai leads CAMPers step by step through screen share on Zoom. CAMPers first created three layers and explored with the tools, including gradients, pencil, fill, etc. 

At the SEC, TAN, and COT groups of the art class, students had a hands-on experience creating abstract art ciphers with the Pigpen Code, the Red/Blue code, and the Crease code. Unlike most of the cryptographic methods we encountered, the pigpen cipher uses symbols instead of letters. Students learned to exchange letters for symbols, which are fragments of a grid. 

(example of a Pigpen Cipher)

This cipher method was used by the Freemasonry and during the American civil war. Even if our encrypted text was hacked, we could always change the key by rearranging the letters in grids. 

Art instructor Tiffany demonstrating the red/blue code

CAMPers also tried the Red/Blue code with our instructor Tiffany Smith. A red/blue system is created when the sender uses a light blue pencil to write their message. Then, they cover that message using red cross-hatching and a highlighter. The sender’s code can later only be revealed using a red piece of cellophane, which makes the light blue writing visible again. 

A Red/Blue code

Last but not least in Tiffany’s art class, we practiced writing a Crease Code. By folding a paper multiple times and placing the secret letters or message on the creases, this technique is somewhat tricky. It requires us to fill the blank space with words that compliment the notes and constitute a complete sentence.  

In the math classroom, Erin Toliver led her class with the COS group. We practiced Modular (clock) Arithmetic, encoding, decoding of Caesar Ciphers with different shifted codes. Tomorrow, we will start pairing students for more encryption and decryption exercises. 

Our CAMPer Marina

Hosted by our TAs: Jazmin, Olivia, Yuxuan, Tsitsi, Alex, Gigi, and CAMPers themselves, the activity periods were filled with diverse projects. Raging from games like Mastermind, Jeopardy!, Zip Zap Zop, SET, to teaching the Binary Number trick, students had a relaxing and fun ending of Day 2. 

For more updates: follow us on Facebook (https://www.facebook.com/BardMathCircle/), Instagram (https://www.instagram.com/bardmathcircle/), and Twitter (https://twitter.com/bardmathcircle)!

CAMP 2020 Day 1- the beginning

It is Day 1 at CAMP 2020, and the first year that we are holding the program online. Since cyberspace shortened the distance between us, the Bard Math Circle received numerous applications from around the country. We see students’ excitement over running into old friends and connecting with new CAMPers in Zoom classrooms. 

We distributed our CAMPers into the six groups: SIN, COS, TAN, SEC, CSC, and COT (trigonometry!) according to their grade level and the questions we sent out on the application form. 

For the morning groups, students and our staff members had an early start of their day during summer. This year, our Math, Computer Science, and Art classes are designed to revolve around the theme cryptography. 

After each group met their instructors, the math class introduced students to vocabularies like plaintext, ciphered or coded text, encode, decode, encrypt, decrypt, etc. The LOWERCASE stands for the plaintext, and the UPPERCASE is the cipher message. Things started to get heated when our teachers brought up a series of mathematical problems related to simple methods of cryptography: substitution, multiplication, keyword cipher, and the period problem. In a substitution cipher, each letter of the alphabet substitutes another letter (e.g., ROT-13: rotate by 13 places). On the other hand, in a multiplicative cipher, the number you multiply plays a crucial role in determining the cipher. 

Table for the times-3 cipher (MULT-3)

If we built a times-3 cipher that multiplies the numbers by 3. As an example, we encrypted the letter c. The number for on the cipher strip is 2, so they multiplied 2 times 3 and got 6. Since 6 is the number for g, they encrypted as G

An encoded message we solved using the multiplication cipher. The plain text is: “To be or not to be, that is a question.”

After a fifteen minute break, we resumed for the art class. The cipher wheel is a circular device using two discs. In lowercase and uppercase letters, the alphabet is recorded on both discs, creating a key to decipher coded messages. CAMP director, Japheth Wood, had mailed the material along with the CAMP t-shirts to the students beforehand. 

A cipher wheel set to key 10

To generate an encoded message using their own cipher wheel, students first need to align the lowercase letter a with any number from 0 to 25 on the center of the wheel. Determining the key (the number) they are using, CAMPers encrypt the plain text according to the matching uppercase letters on the inner disc. Noah in the SIN group sent his classmates the following text:

ROVVY 10

According to his key: the number 10, we figured out that he said “HELLO.” The class also decoded a few messages pre-designed by the instructors. Chelsea, a senior instructor for the art section, shared her screen for a long encoded message. We notice that the letters K, Y, V are repeating themselves: these letters stand for “the” in her sentence: “the unicorn is the national animal of Scotland.”

“The unicorn is the national animal of Scotland.”

During the CS class, students learned some basic necessary commands on Google Colaboratory: ord, chr, append, join, print, for/in, if/else, return, def, print, strings, lists. Further, CAMPers explored how each command works and proposed various questions on how to change the command on their copy of the shared document. 

Approaching the end of day one, our junior staff planned various activities for their group tailored to students’ interests. Some led the classroom to further explore the topic they left off, and some invited CAMPers to play the SET game as a tradition of our program. 

We had a productive first day at CAMP 2020. Stay tuned for more updates on Facebook (https://www.facebook.com/BardMathCircle/), Instagram (https://www.instagram.com/bardmathcircle/), and Twitter (https://twitter.com/bardmathcircle)! See you tomorrow.