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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. 

“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.

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!