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We have officially reached the end of Bard Math CAMP 2025! This week has really flown by, occupied by great fun as well as great math. Students enter the lobby and make their way to morning activities to pass time as they wait for their last math class of the week.

The Sine group starts by exploring a grid, noticing how to get from one intersection point to another, recognizing a pattern and making it easy to traverse from “Point A” to “Point B”. They discover Pascal’s Triangle within the pattern!

CAMPers then use exactly three Cuisenaire Rods to see how many different trains could be made for a set of sequential numbers. The CAMPers collaboratively fill out a data table on the board, taking turns each adding a component. They discuss patterns between length and number of trains. CAMPers see how “choosing” played a role in finding the numbers and the “magical” appearance of the triangular numbers.

Class ends with CAMPers having the opportunity to create even more trains, with each group getting a different criteria. Unfortunately, time runs out to finish this activity- but with enough information to continue on their own!

The Cosine group starts class going over the methods of solving the train problem they were given yesterday, counting the number of trains of a certain length, composed of a certain number of cars. CAMPers are then tasked with explaining what those lengths mean in the context of a formula that one CAMPer conjectured: The number of trains of length n made from k cars is just (n-1) choose (k-1)!

They use blocks to create a visual representation of the trains in the problem and are challenged to use these visuals to investigate correlations between the numbers in the problem and triangular numbers. Class ends with CAMPers visiting each others tables, taking turns explaining patterns and thought processes to solve the problem.

CAMPers then move on to their second class of the day with the Cotangent group heading to art and the Tangent group going to computer science.

In art, using the vanishing point and perspective concepts that they learned earlier this week combined with their shading practice yesterday, CAMPers make optical illusion-like doodles! They create squiggly oceans, descending rabbit holes, colorful patterns, and so much more.

In computer science, the Tangent group starts by collaboratively completing the Towers of Hanoi. Each CAMPer is assigned a disk and is responsible for said disk and the one above it. The goal is to discover a pattern of behavior that will be translated into a recursive algorithm!

The CAMPers then use the driver-navigator partner system to write a code that efficiently solves the Towers of Hanoi using the patterns they explored and their newfound understanding of recursive functions.

CAMPers break for lunch, enjoying the beautiful weather! Skipping our elective period, once CAMPers are done with lunch, they head straight to their third class of the day. The Cotangent group goes to computer science while the Tangent group heads to art to have their turn in creating and designing optical illusions.

In computer science the Cotangent group gets a new challenge to work on! Some are continuing work on logic problems like determining whether inputs are positive or negative. Others use the driver-navigator partner system to create a function that doesn’t use loops, but finds the factorial of a number and test their thought processes in this.

When CAMPers finish this challenge, they are given the same Tower of Hanoi problem that was given to the Tangent group. They work to figure out a pattern so they can create a function that will demonstrate it, and solve this puzzle.

After their last class at Bard Math CAMP, CAMPers take a survey asking their opinions on the time they spent here. They trade their survey in for their CAMP t-shirt! All CAMPers walk to the RKC building to take a photo together. Once back, everyone sets up and gets ready to show their parents everything they’ve worked on this week during open house.

We have officially reached the end of Bard Math CAMP 2025! We hope that all CAMPers were able to learn something new and will continue exploring any curiosities outside of CAMP. It is such a joy to see so many CAMPers grow intellectually while having so much fun doing math!

This week went by so fast! Although we hate to see this week go, saying goodbye means enthusiasm for next year just grows, as more ideas for what to do next continues to form!

Photo credits: Featured image- Bard student on campus, 1-39- Sabine Harvey

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!