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Welcome to the final day of Bard Math CAMP 2024!

The day kicked off with a memorable moment as all CAMPers gathered in their Bard Math CAMP 2024 T-shirts for a group photo. This snapshot captured the spirit and camaraderie that had built up over the week, creating a lasting memory for everyone involved. The excitement was palpable as everyone beamed with pride, commemorating their week of hard work and learning. After the photo, CAMPers filed upstairs for their final Math Classes of the week.

In Japheth’s Cosine Math Class, CAMPers dove head first into the world of star polygons. They completed their scatter projects, showcasing their understanding of complex mathematical concepts through creative presentations. Once finished, they moved on to creating their own star polygons, some expanding the ideas of what a star polygon could be, honing their creative and geometric boundaries.

The class also delved into the inscribed angles theorem, with a special focus on an isosceles triangle proof guided by Mason. This blend of hands-on activities and theoretical learning made for an enriching final session. The challenge of constructing massive star polygons fostered teamwork and problem-solving skills among CAMPers, leaving them with a profound sense of accomplishment.

Frances’ Sine Math Class began with cube nets, challenging CAMPers to hone their visualization skills and mentally determine which nets could form cubes. This exercise sharpened their spatial reasoning abilities and deepened their understanding of geometric properties. The CAMPers then used Zometools to create Archimedean polyhedra, constructing more complicated 3D models. This activity not only tested their geometric knowledge but also their patience and teamwork. As they built more models, they learned how adding faces impacts the angle degrees, further deepening their understanding of geometric principles.

In the afternoon, CAMPers had lunch provided by Parkhurst Dining, and spent their afternoons in a variety of different electives:

CAMPers continued exploring breadboards in Intro to Circuits, gaining confidence in their ability to explain how they worked to Yulia and Shaylen. This activity reinforced their understanding and comfort with circuitry concepts. They also experimented with more complex circuit designs, learning about the impact of resistance and voltage on their projects.

In Art Class, CAMPers created fun photo booth props for their group photos, adding a playful touch to their final day memories. They designed and decorated props that reflected their individual personalities and experiences from the week, making the photo session even more enjoyable.

Frances guided CAMPers back to the Zometools in her Polyhedra Class to create even more complex polyhedra for the Open House. This session allowed CAMPers to showcase the advanced geometric models they had learned to construct. They marveled at the intricate structures they were able to build, applying their knowledge of Euler’s formula and other geometric principles.

In Shoshi’s Art Class, CAMPers busied themselves with finishing projects from the week. They put the final touches on their circuits, completed the massive magic cube, and crafted intricate origami pieces for the Open House. This creative flurry ensured that all their hard work was ready to be showcased to family and friends. The atmosphere was one of joyful collaboration, as CAMPers shared ideas and techniques, enhancing each other’s projects. Shoshi’s guidance helped them see the connections between art and mathematics, highlighting the beauty of geometric design in their creations.

Yulia’s Computer Science Class saw CAMPers applying all the skills they had learned throughout the week to create intricate tessellations with NetLogo. This final project allowed them to demonstrate their coding prowess and creativity, culminating in beautiful, mathematically complex designs. CAMPers explored the use of sliders and repeat commands to automate the creation of polygons, enhancing the complexity of their tessellations. Yulia’s encouragement and expertise helped them push the boundaries of their coding abilities, resulting in impressive and unique patterns.

At 3:00PM, the Bard Math CAMP team was thrilled to welcome families to the Open House, where they had the opportunity to revel in the CAMPers’ hard work and accomplishments from the week. As families explored the various displays, they were greeted by a vibrant showcase of intricate tessellations, complex polyhedra, slideshows, and creative art projects. The pride and excitement on the CAMPers’ faces were matched by the admiration and amazement of their families. It was a joyous occasion, filled with enthusiastic explanations, demonstrations, and heartfelt moments of recognition. The team was immensely proud to see the CAMPers confidently presenting their projects and sharing the knowledge and skills they had gained.

This celebration of learning and creativity was the perfect culmination of a remarkable week at Bard Math CAMP. From learning complex mathematical theories to creating intricate art projects and building lasting friendships, Bard Math CAMP 2024 was a resounding success. The CAMPers’ enthusiasm, curiosity, and determination made this week truly special. We hope the skills and memories forged here will inspire these young minds for years to come. Thank you for being a part of this extraordinary experience! We look forward to seeing what our CAMPers will achieve in the future, armed with the knowledge and confidence they gained this week.

It was a pleasure to blog for you all – have a great rest of your summer! – Jaelyn Quilizapa

Welcome back to another Bard Math CAMP Blog!

Here’s a detailed look at today’s engaging activities and lessons:

The day began with energy as CAMPers engaged in a variety of opening activities, such as games, tessallations, and art projects, including one giant magic cube! These activities set a positive start for the rest of the day, preparing the CAMPers for the lessons and explorations ahead.

In Frances’ Sine class, the day started with a review of the two methods CAMPers used to determine the number of degrees in each angle of a regular polygon. They discovered that, when written in algebraic notation, both methods yielded the same results, reinforcing their understanding of mathematical consistency. Next, they moved on to constructing polyhedra with Magnatiles. This hands-on activity helped them explore the relationships between the number of faces, edges, and vertices in these three-dimensional shapes.

CAMPers then delved into the properties of regular polyhedra. CAMPers began building these fascinating shapes using Zometools, which provided a more precise and intricate method for constructing polyhedra. This activity allowed them to see firsthand how the geometric properties they had been studying came together in complex, three-dimensional forms.

In Japheth’s Cosine class, CAMPers explored the fundamental differences between polyhedra and polygons. They learned about convex shapes and how these relate to Platonic solids, which are the building blocks of many complex geometric forms. The class examined 20 different nets of a cube, challenging CAMPers to mentally determine which ones could form a proper cube without physically assembling them. This exercise significantly improved their mental imagery skills.

Additionally, CAMPers made their own cube nets without lifting their pens, requiring them to plan their moves carefully before executing them. The class also explored polyhedra algebraically using inequalities and discussed the concepts of nets, which provided a deeper understanding of geometric structures.

Today’s lunch was once again provided by Tastebudds, and with the beautiful weather, CAMPers took the opportunity to sit outside and enjoy their midday break.

As the afternoon rolled in, CAMPers eagerly transitioned to their elective activities, where they could explore a variety of interests and hands-on projects. Here’s a look at what each elective had to offer today:

CAMPers were thrilled as they learned new math magic tricks from Frances, which they performed for their peers. This elective, though magical, are solely based on mathematical principles, creating an exciting and educational experience.

Intro to Circuitry was a blend of theory and practical application. CAMPers reviewed circuit symbols and experimented with breadboards, enhancing their understanding of electronic components. Shiven explained electromagnetic fields around magnets and related this to the functionality of a hand-crank flashlight, helping CAMPers grasp the principles of electricity conduction. They continued their projects from previous days, but now conducting electricity through a breadboard in order to make their LED bulbs light up.

In the Origami elective, CAMPers created Lilies and Hummingbirds with Ariella!

Shoshi’s Art Class was a creative haven where CAMPers designed and created 3D cubes using their own dimensions. This hands-on activity allowed them to apply their understanding of geometry in a tangible way. CAMPers personalized their cubes with unique designs, colors, and patterns!

Yulia’s Computer Science class was a hub of activity as CAMPers delved into more advanced programming concepts. They learned about sliders and the repeat command, which allowed them to set the number of sides and the length of each side, making their tessellations more intricate and complex. CAMPers learned to create buttons that could automatically generate polygons, streamlining their coding process and making it more efficient. This, paired with revising their triangle codes from the previous day, taught them valuable coding techniques but also showed them how to apply these skills to create beautiful, mathematically accurate designs.

For closing activities, some CAMPers continued with their tessellations, magic cubes, and origami projects, while others tackled Mason’s Math Challenge!

Along with this, CAMPers were asked, “What made you smile today?”, their answers are as follows:

As Day 4 draws to a close, the excitement and learning at the Bard Math CAMP continues to build. Each activity and lesson contributes to the CAMPers’ growing knowledge and enthusiasm. We can’t wait to see families and CAMPers back for the last day of CAMP!

By Wednesday, we’ve reached the very middle of the Math CAMP week.  So far, the CAMPers have explored a wide range of math puzzles and games, as well as having learned about binary (writing numbers in Base 2, as strings of 0’s and 1’s).  This morning, we had a special guest lecture by math professor Charles Doran, which taught us the mathematics behind the Towers of Hanoi.

• What is a Graph? – We opened with a discussion of the Konigsberg Bridge problem (see images below): Is it possible to take a walk through town, crossing over each bridge exactly once? By shrinking down each landmass to a point (vertex), because we don’t care about shape or size, and by making each bridge a line (edge) connecting two vertices, we can determine whether or not there exists a path through every edge, with no repeats. This object is what we call a graph. But what if we want to find a path that goes through each vertex exactly once?
• Hamiltonian Paths – That last problem is what an Irish mathematician named William Rowan Hamilton (not to be confused with founding father Alexander Hamilton!) wanted to find the answer to. The CAMPers were treated to a brief sample of an A Capella Science Hamilton parody video (William Rowan Hamilton (Science YouTuber Collab) | A Capella Science – YouTube) before taking a look at a set of graphs Qn (n = 1, 2, 3, …): Q1 consisting of 2 vertices connected by 1 edge (drawn as a line segment), Q2 having 4 vertices connected by 4 edges (drawn as a square), Q3 being drawn as a cube in 3-space.

• Naming Vertices – So now we have a good set of graphs to work with, but is there a way to find a Hamiltonian Path (one that passes through each vertex once) without paying any attention to the edges, since the vertices are the only thing we care about? Using coordinates to name the vertices (0 and 1 on Q1, x- and y- coordinates on Q2, and x-, y-, and z-coordinates on Q3), we can see that the pairs of vertices connected by one edge are those that use one bit-flip (a switch from 0 to 1, or vice-versa), and paths are a sequence of vertices such that every pair is connected by 1 edge. So, a path that goes through all the vertices is really a sequence of vertices whose x-y-z coordinates only change by one bit-flip each time.
• Gray Codes – We tried to write the sequence of vertices on the cube as numbers 1-7 in Base 10, converted into binary (Base 2), but we soon realized that there were too many steps involving more than one bit-flip, which doesn’t make a path. To fix this, we were given two recipes for the conversion binary numbers to the Gray Code system, which gives us a perfect Hamiltonian Path (exactly one bit-flip to take us from each vertex to the next).

• Baguenaudier Puzzle – One of the best things about Gray Codes is that they can help us solve math puzzles – from Baguenaudier (“time-waster”) rings (The Chinese Rings Puzzle (wolframcloud.com)) to, you guessed it, the Towers of Hanoi.
• Towers of Hanoi – Many of our CAMPers have already solved the six-ring version of the Towers of Hanoi, but with Gray Codes we can figure out the solution to the n-ring version of the puzzle – no matter how many rings there are, the Gray Codes never fail. All we have to do is interpret each bit-flip as a transfer of one ring to a different tower (and since there are n towers, our Gray Code numbers will have n places). However, the fun’s not over yet – How can we use Gray Codes to solve the puzzle when someone hands us a partially completed Tower of Hanoi? This was one of the several new questions that we were left with at the end of the lecture.

After the lecture, CAMPers went off to Math class, where those in the COS group continued working with the Cantor Set, starting off by brainstorming a list of deceptively simple questions – Is 1/4 in the Cantor Set? What about 3/4? Is 0.9999… the same thing as 1?. Then they started using algebra to convert numbers in different bases (e.g., 0.2020202020…, which is in Base 4) into fractions in simplest form.

The SINE group continued to explore the properties of the Sierpinski Triangle, this time working to find its dimension (at first glance, it looks 2-dimensional… but is it, really?), beginning by finding the scaling factor.

In CS class, the SEC group built off of what they had learned about binary and Gray Code numbers in the guest lecture, continuing to break down place value in Base 10, which led them to a convenient way of converting binary numbers back to decimal (Base 10) – multiply the 0 or 1 in each place with powers of the base (in this case, 2) which indicate the place value of the digit (e.g., 110 = (1 x 2^2) + (1 x 2^1) + (0 x 2^0) = (1 x 4) + (1 x 2) + (0) = 4 + 2 + 0 = 6, and you’ll get the same number in Base 10).

After having lunch at Kline Dining Commons, CAMPers chose between playing Conway’s Game of Life, taking a hike to the Parliament of Reality, and the decoration of another mysterious wire sculpture – this one, tucked behind the Chapel of Holy Innocents, is even more abandoned than the one we decorated on Day 2, complete with just-as-intricate cobwebs stretching between the metal rods. (This elective also included a very intense sponge race.)

Once the electives were over, CAMPers in SEC went to Art class – where they continued making paper cubes.

In the last half hour of CAMP, the groups reconvened in the Auditorium (RKC) for more math activities – Rubik’s Cubes, Hex, an M. C. Escher memory game, and (naturally) the Towers of Hanoi.

Day 3 started off with puzzles, segued into the binary system, explored irregular objects with unusual dimensions, and ended with more puzzles. We’re eager to find out what math will await us on Thursday morning!

Photo Credit: Sonita Alizada (images 21-25, featured image), Kateri Doran (images 1-8, 10-20, 26-35), screenshot by Kateri Doran (image 9).

The second day of CAMP this year is off to an excellent start – armed with their new knowledge of logic and truth, the CAMPers are ready to begin their exciting journey into the magical world of fractals. 

We started off the day with a whole new set of math puzzles – hypothetical chocolate boxes, tangrams, Rush Hour, linking puzzles, and of course, Set. 

“… You take a card, and you put down the cars [that are] on the card, and you have to get the red car out by only sliding the cars this way.”

“So basically, we lay out cards, and they all have different patterns, and you [want to] find the sets of 3.”

“So basically, there are clues that you get in the 4-by-4 box, and you have to figure out based on these clues the color and shape of the chocolate.”

The COS group began by breaking into pairs and working on a sheet of logic problems. After that, they went on to review truth tables before Japheth switched gears to talk about fractals.

• Sierpinski Triangle – Starting with one equilateral triangle, we break the triangle into four smaller triangles and take out the triangle in the middle. Now we have three smaller triangles. Now break each of those into four even smaller triangles – then take the middle triangle out…
• Tree/”Neuron” Fractal – Start with a line that branches out into a “Y” shape. Now branch each of those smaller branches into two “Y” shapes. Now branch each of those…
• Fibonacci Spiral – Believe it or not, the Bard Math Circle logo is actually a fractal! It starts with the smallest rectangle and doesn’t get any smaller – it gets bigger. If you start with a rectangle with side-length ratio 1/2, then make a bigger rectangle by adding a rectangle with ratio 2, then continuing to add on rectangles with a bigger ratio each time (following the numbers in the Fibonacci sequence [1, 1, 2, 3, 5, 8, 13…]), then draw a curve through each of the rectangles… you get a Fibonacci Spiral.
• Cantor Set – Starting with the space between 0 and 1 on the number line, get rid of the interval between 0.333333… (1/3) and 0.666666… (2/3). Now we have two intervals [0, 1/3] and [2/3, 1]. Now take the middle third from those. Now we have two more intervals. Now take the middle third from those…

Meanwhile, SINE continued their work with using logical operators to write out true (or false) statements, which led to several productive debates over what the statements really mean, beginning with contention over compound statements (What happens when you negate a statement that already includes an AND or an OR, e.g. NOT [c OR e] ?).

After that, they segued into an exploration of the perimeter and area of the Sierpinski Triangle at various stages, using it as an example of a fractal with self-similarity – that is, if you take a small copy of the original fractal and expand it, it will look exactly the same as the original, with smaller and smaller and infinitely smaller triangles trapped inside of it.

Next, the SEC group crossed the building to join CS class, while CSC enjoyed a sunny stroll to Hegeman for Art, where they were given business cards to fold into paper cubes.

In CS, the CAMPers were introduced to a new logical operator, XOR (exclusive or) – the output only true if one (not both; only one) of the inputs is true. They then split into two groups to try and create an actual, physical XOR statement with batteries, LEDs, and cables.

Next, they transitioned into an introduction to the Base 2 (binary) system, travelling back in time to the days when place value in Base 10 (e.g., 4,598 = 4 thousands, 5 hundreds, 9 tens, and 8 ones) was heavily emphasized in their 2nd-grade math classes.

After a lunch at DTR (Down the Road), several heaping platefuls of cookies, and a not-so-secret trip to the bookstore, CAMPers split into groups based on the electives they had signed up for at the beginning of the day – a hike to the Blithewood garden, the ever-popular billiards table, and the “do-decoration” of a very mysterious dodecahedron.

No one on Bard campus can remember when the wire sculpture arrived – and even those who remember a time before can’t recall where it came from, or who put it there.  Nevertheless, the rusty, dilapidated dodecahedron has been exposed to the elements for decades at least – and it blends in so well that most people who pass by need to squint to see it clearly.  CAMP has decided to change that. 

At the end of the electives period, CAMPers in CSC and SEC returned to RKC and Hegeman, respectively, for their CS and Art classes. Finally, everyone got together for the end-of-day activities – river-crossing puzzles, Rubik’s cubes, more linking puzzles, and Hex.

Day 2 of CAMP was full of color, laughter, and lots of fractals – we’re excited to see what Day 3 will bring!

Photo Credit: Sonita Alizada (images 7-8, 10-14, 22-32), Kateri Doran (images 6, 9, 15-17, 33), Shiven Dabhi (image 18), Japheth Wood (images 19-21), public domain (images 1-5)

It’s the first day of the 9th year of Bard Math CAMP – the first day in two years that our CAMPers and instructors have been able to do math in three dimensions!  In other words, this is the start of CAMP’s first in-person session after two years of meeting online.  

First thing in the morning, the students were treated to an indoor math fair of sorts – the first two rows of the László Z. Bitó auditorium were decked with math puzzles, fractal magnets, and the board game Set.

“We basically have to figure out the size of each skyscraper – like, from here you’re going to have to see two skyscrapers, but you can’t have the same number in each row.”

“This is a deep-sea diver, and this is a mermaid. … The human race is corrupting all beings, even in the ocean, and soon we will not be able to escape the humans.”

In Frances’ math class, the CAMPers in the SINE group were shipwrecked on a mysterious island…

You land on an island where the people either always tell the truth or always lie.  You need to get your bearings on this island, and you see three islanders walking up to you.

You ask the first person: “Are you a truth-teller?”  But you don’t hear what he says.  

The second person says that Person 1 said “yes”.  

The third person says that Person 2 is a liar.    

From this, they went on to learn about logical operators (NOT, AND, OR) in preparation for Computer Science (CS) class.  

In Art class, students in the CSC group were given a piece of construction paper and, making a series of rectangular, triangular, or circular cuts along its folded edge, were tasked with making their 3-D design as intricate as possible.

In CS class, the SEC group continued their work in logic with truth tables.

Setting the two inputs (A and B) to either 0 or 1, CAMPers used their new knowledge of logical operators to determine whether a given statement (the output) was true or false.

The CAMPers continued to explore inputs and outputs by connecting batteries (inputs of energy) and LEDs (outputs of light) via cables or splitters to create “statements” in real life.

After having lunch at Down the Road Café, CAMPers had the option to either linger in the Campus Center and try their hand at billiards/foosball, to follow Japheth on a hike to the Sawkill Stream, or to head back to the Reem-Kayden Center and learn magic tricks with Frances.

After that, the SEC students headed to Art class for mathematical crafting, while the CSC students went to the RKC computer lab to learn more about logic and truth tables – engaging in a somewhat philosophical discussion about why computers use 1’s and 0’s, and why the truth or falsity of statements in CS class is not at all up to interpretation.

At the end of the day, the groups converged once again in the Bitó Auditorium for more math activities – origami, card tricks, and the Game of Hex.

The first day of CAMP was exciting and eventful for everyone involved – we can’t wait for more math adventures on Day 2!