Calling all CAMPers, parents, and curious readers – CAMP math instructors Japheth and Frances have a few fun problems for you to think about for the rest of the summer:
Cookies and Kids: Kids A, B, C share a certain number of cookies (some might get zero). Find a systematic way to represent each way to share. Then arrange all the ways to share in a triangle. Hint: This problem has a hidden relationship involving the numbers 2 and 5. Can you find it? Bonus Question: Can you find a pattern for different numbers of kids, and/or different numbers of cookies?
Grid Paths: Point B is two blocks East and some number of blocks South from point A. How many paths are there from point A to point B that are of the shortest possible length? Find a systematic way to represent each possible path. Then arrange all the paths in a triangle. Bonus Question: We also looked at shortest walking paths in a square grid with walks North and East. Is there a pattern to the number of possible shortest walking paths when walking 2 blocks North and b blocks East? What about when walking 3 blocks North and b blocks East?
Spot It! : Seven was the largest number of symbols and cards we found if there are 3 symbols per card. Is 7 the largest possible number of cards that could be made with 3 symbols per card? Prove (or disprove) this statement. Hint: How can we know when something is impossible or possible but not yet figured out?
Spot It! – Bonus Question 1: Is the largest number of cards that can be made always equal to the number of symbols? Bonus Question 2: Can decks of Spot It! Cards be made for any number of symbols per card or are there numbers for which cards cannot be created? Bonus Question 3: The decks sold have missing cards. Are there efficient ways to find which cards were omitted? Bonus Question 4: How many cards and symbols are there in a deck of 7 Spot It? Then design a deck of 7-Spot It, which some people claim does not exist.
Fractions: 2/3 x 5/7 can be thought of as “2/3 of 5/7”. If we use a circle to represent the number one, what can be drawn in the circle to demonstrate the meaning of and answer to 2 ⁄ 3 x 5/7 ?
Pascal’s Triangle: Pascal’s Triangle is full of interesting relationships. Keep looking at it and write the ones you discover. What other questions do you have about these topics?
It’s the 5th and final day of the 10th year of CAMP. This morning, while they puzzled out Slitherlink challenges, magic tricks, and Set – games that many of them had never seen before the beginning of the week – the CAMPers recieved their ocean blue T-shirts, just in time for a big group photo on the stairs inside RKC.
After that, the CAMPers split into SINE and COS for math class, where both groups were working on ice cream bowl problems (and their variations). The SINE group game together to discuss their solutions to the problems they worked on yesterday: the ice cream problem, the handshake problem, the shortest-paths problem, and the sharing-cookies problem. Next, Frances introduced them to Pascal’s Triangle, a triangle of numbers where the middle entry below two numbers is always the sum of those two numbers. The CAMPers constructed the triangle from scratch, then looked along its diagonals, finding the counting numbers (1, 2, 3, 4, 5, etc) and the triangular numbers.
“In France, there are actually streets named after mathematicians and schools named after mathematicians… There’s a street named after Pascal.”
– Frances
Meanwhile, COS continued working on their table from Day 4, which showed the number of ice cream flavors and the number of ice cream bowl (ICB) combinations for a 2-scoop ICB. The CAMPers observed that the number of ICB combinations was always a triangular number (a number made from summing up counting numbers): 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, 1 + 2+ 3 + 4 + 5 = 15, etc … . Japheth then handed out individual problems for each of the smaller groups to investigate. Their goal was to find an isomorphism between them – to figure out what made the problems the same. They found that the handshake problem was isomorphic to (the same as) the ice cream bowl problem… the two remaining problems – shortest paths and cookies – were left to the CAMPers as puzzles to think about for the rest of the summer.
In Art class, the CSC group wrapped up their Spot It! decks, origami cranes, and doodles for the end-of-CAMP photo booth.
Meanwhile, in CS class, the SEC group worked on drawing a ball, making it move, and controlling all the aspects of its motion on the canvas – for example, speed in a certain direction, with ballspeed_x and ballspeed_y. Next, they learned about void draw(), void setup(), and void keyPressed(), which helps to organize the code, making it look less messy on the screen by grouping pieces of code together. At the end of class, Arnav showed the CAMPers some of the mesmerizing creations that use Java processing – for example, https://bleuje.com/gifanimationsite/2023_1/ .
“You should all be proud of yourselves… in, like, 5 hours of study, you’ve already covered more in processing than a lot of people have in their college classes.”
– Arnav
After class, CAMPers grouped together for Friday electives – AMC contest problems, decorating the photo booth, and more tic-tac-toe variations with sidewalk chalk.
In Computer Science, the CSC group learned more about the inner workings of a computer. They talked about input – the information that the computer receives, from buttons on a keyboard or a microphone – and output, which appears on the screen or from the speakers. The input is stored as the computer’s memory, either volatile (temporary) or non-volatile (permanent, unless you delete it). They talked about the CPU (Central Processing Unit) before switching gears to talk about text binary – the CAMPers wrote “hello” notes to each other in 0’s and 1’s as their last activity before the open house…
… where, at the end of the day, the CAMPers were given the chance to showcase the puzzles, games, and code they had been working on all week.
(A Tip for Parents: If you want a first-hand account of what your child has been learning this week, be sure to take a peek at their notebook! )
The 10th year of CAMP was an absolute blast – an explosion of creativity, where everyone jumped in to ask questions, try ideas out, and work as a team on a quest for the answer. This week, the CAMPers dove right into new concepts, exploring areas of math, CS, and art that most of them had never touched before – from magic tricks to juggling to coding in Java, to the intricacies of ice cream scoops and Spot It! and computers. Wherever they go next, they’ll be able to tap into the new kind of thinking they learned at CAMP – a mindset of questioning, exploration, and discovery.
The sun is shining over RKC – setting the stage for lots of summer fun at CAMP! This morning, CAMPers got together in the auditorium to solve more math puzzles – the Towers of Hanoi, Rush Hour, and shape-matching challenges. The problem of the day: Completely fill a 3×3 grid with X’s and O’s. How many of these configurations have three X’s or O’s in a row?
In math class today, SINE broke into four smaller groups to work on four different but similar problems: finding the number of possible handshakes between 5 people, then taking 3-way handshakes into account; finding the number of 2-scoop ice cream bowls you can make with 5 different flavors, then doing the same for 3-scoop bowls; figuring out the shortest path out of all the different ways to move by 2 blocks north and 3 blocks east, then generalizing to b blocks in any direction; and finally, determining the number of ways for 3 kids to share 3 cookies – or more than 3 cookies.
“The ice cream bowls are shaking hands!”
Meanwhile, the COS group started analyzing 8-Spot-It! – the classic version. They started with existing Spot It! decks, putting cards in piles based on the symbols they found on a randomly-drawn card C. Next, they chose one pile (A) to put back in the box, laying out the remaining cards in columns and rows that matched the top card of the pile, and discussed the observations and questions they had.
After that, they switched gears to solving the same ice cream bowl problem that the SINE group was working on – How many unique 3-scoop ice cream bowls can we get with 5 different flavors of ice cream? The first scoop gave them 5 choices, the second 4 choices, and the third 3 choices, which multiplied together to give 60 choices. However, the CAMPers found that in counting combinations they had actually over-counted by a factor of 6, since there were 6 ways to get the same ice cream bowl with each combination of 3 flavors.
After math class, CAMPers in the CSC group walked over to the art room, where they worked on origami, highly ambitious 7-symbol Spot It! decks, and beaded Fibonacci necklaces.
In Computer Science, the SEC group learned about using booleans (true or false) and comparators (>, <, greater than or equal to, less than or equal to) to write conditionals: “if” statements with conditions on variables x and y. They also learned about “if else” statements, which tell the computer what to do if the variable does not satisfy the condition. Breaking off into their usual groups, the CAMPers used these concepts to make their drawings bounce in different directions – up and down, side to side, and diagonally.
The CAMPers had lots of fun outdoor electives to choose from – a hike to the Sawkill waterfall, more magic tricks with Frances, variations on tic-tac-toe with sidewalk chalk outside the building, and, drumroll please: Break-Your-Brain Rhythm Games!
After lunch, the CSC group learned about bits and bytes, which are used to store data in a computer – a bit takes a value of 1 or 0, while a byte is an 8-bit (binary) number. The instructors explained why kilobytes, megabytes, gigabytes, and terabytes aren’t “clean” numbers ending in a bunch of 0’s: All of them are powers of 2.
The CAMPers were treated to a special investigation – opening up an actual computer and taking it apart! This allowed them to search for the basic parts of the computer and answer questions: Where does it hold its power? Where does the memory live?
The CAMPers were able to find….
The Central Processing Unit (CPU) …
The transistors, little switches that flip between 0 and 1 …
The motherboard, the most important circuit board, which basically holds everything together …
The RAM (Random Access Memory) chips, which store the computer’s volatile memory …
The capacitors, which are like little battery tanks that hold small amounts of energy …
The hard drive, which is made up of spinning magnetic disks that encode 0’s and 1’s …
… and the fans (to keep the CPU from overheating).
At the end of a long and eventful day, the CAMPers wound down by playing math games and working on their magic tricks.
Day 4 was full of new adventures, breathtaking views, and mathematical magic. We can’t wait to find out what tomorrow will bring!
This morning, the CAMPers gathered in the auditorium for a collection of math puzzles and the new problem of the day: How many Tower of Hanoi board states are there? The CAMPers had fun trying to work it out, rewriting the problem on the board – in the Greek alphabet. Right before breaking up into their usual Math groups, Japheth led them all in the InternationalMath Salute.
Following up on the argument they made yesterday, the COS group worked together to prove that there couldn’t be an 8th symbol in a 3-Spot-It! deck. They then drew up a graph that brings to mind the “Deathly Hallows” from Harry Potter, showing the 7 cards as nodes and the symbol in common between two cards as edges. Next, they broke up into smaller groups to work on designing a 4-Spot-It! deck – What’s the greatest number of cards we can get with 4 symbols on each card? Some of the CAMPers observed that for the 2- , 3- , and 4-Spot-It! cases, the number of symbols was the same as the number of cards. To look for more patterns, they took the difference between the number of cards in n-Spot-It! and (n-1)-Spot-It! , then took the difference between the differences – getting 2 every time. Leaving this conjecture as a cliffhanger for a later date, the CAMPers embarked on a final investigation – to make one pile for each symbol using existing Spot It! decks.
Meanwhile, the SINE group used tree diagrams to explore the main concepts of combinatorics. Taking the letters S, T, R, and E, they mapped out all the possibilities of what letter could come after each letter in a word, branches branching off into other branches. They followed different branches to make different words like SET, REST, STREET, TEETERS, and – appropriately – TREES. The CAMPers continued their discussion by breaking into two smaller groups to work on these examples: choosing ice cream flavors for a multi-scoop cone, and choosing paths to take when going from Point A to Point B. How many combinations are there? How many choices do we have?
In Computer Science class, the SEC group learned about the random() function, void setup() and void() , variable assignment – how to turn x into (x+1) in order to move a shape somewhere else on the canvas. Their challenge for today: Rather than just moving your drawing around, can you make your drawing move in an animation?
After class, the students joined their elective groups: more magic tricks, more paper puzzles (AKA the Harry Potter debate club), as well as origami.
During lunch, the rain started pouring outside Kline, and the CAMPers had to break out their umbrellas to make it to their next classes – Art and Computer Science.
CSC went to Computer Science, where they continued to learn about truth tables. Then they broke into small groups, using logical operators to write out a combination of A and B that was equivalent to the XOR (exclusive or) operator. Then they took out Little Bits (batteries, cables, and LEDs) to make an XOR statement in real life.
In the Art room, music was blasting while the SEC group continued working on their Spot It! cards, then dove into a variety of artistic and mathematical endeavors – Fibonacci challenges, 4×4 tic-tac-toe, pink paper airplanes,and optical illusions.
Afterwards, the two groups converged in the auditorium – finally dry and ready for some math games.
New ideas, new methods, new questions and answers filled Day 3 with color, excitement, and fun! The CAMPers made so many new discoveries today, leading to even more unanswered questions – perfect to tackle with zeal on Day 4.
The second day of this year’s CAMP started off with a variety of fun math games – Ghost Blitz, Spot It! , a big game of Set, the towers of Hanoi, and a miniature version of chess. The CAMPers were also presented with a “problem of the day”: How many possible Ghost Blitz cards are there?
“… If the object is on the card, then you grab [that object], and if not, you grab the [object] that has nothing in common with the others.”
In Frances’ math class, small groups within SINE had come up with different conclusions about how many cards you could possibly get with a certain number symbols on each card. They were each asked to draw and explain their diagram solutions at the board – some using the “line method”, some using the “grouping method” to show every combination of symbols.
Meanwhile, the COS group launched into an in-depth analysis of Spot It!(both the full and junior versions), making observations to ask questions about how the game works and trying it out together as a class by constructing simple versions – 0 symbols, 1 symbol, 2 symbols – from scratch. Next, the broke into small groups to tackle the 3-symbol case.
To get started, one group decided to use digits instead of letters to stand in for symbols, recognizing that Rule #3 of Spot It!caused Symbol 1 to appear on every card. They then used process of elimination to figure out what the other digits (symbols) on each card should be – once two cards have one symbol in common, they can’t have any other symbols in common. By the end of class, the CAMPers had come up with apretty solid argumentthat with 3 symbols per card, there must be 7 cards.
After math class, the SEC group brought their flash drives to Computer Science, where they continued their exploration of shapes and coordinates, then brainstormed how they would go about making shapes that change. Next, they talked about how to fill in color values and how RGB colors channels work – each 24-bit pixel is split between the R channel, the G channel, and the B channel, so they each get 8 pixels. After that, they split into groups, planning out the code to make a shape and change its color – on paper, before they tackled the computers.
After class, CAMPers met in the RKC lobby to sort themselves into their pre-lunch elective groups: learning mathematical magic tricks with Frances, practicing the ability to think fast with improv games, and solving more pencil-and-paper puzzles.
After a great lunch at the Kline Dining Commons (plus unlimited scoops of delicious local ice cream), the CAMPers split up again into CSC and SEC for Computer Science and Art, respectively.
Computer Science at CAMP is different from the kinds of “coding classes” that many middle-school students take online. The CAMP instructors choose to focus on the underlying properties of computers and the instructions that humans feed into them. Understanding binary (strings of 0’s and 1’s that encode base-10 numbers), as well as logical operators (NOT, AND, OR, etc.), sets the foundation for the CAMPers to learn any programming language they choose in the future. With those two concepts in mind, the CSC group started working with truth tables to put it all together.
Meanwhile, the SEC group made progress on their Spot It! decks, adding the symbols they designed to index cards.
At the end of the day, the CAMPers all filed into the Bito Auditorium for more math games, like Ghost Blitz, Chocolate Fix, Rush Hour, and Set.
Day 2 got all the CAMPers thinking deeper and going further, asking questions and working together to find the answers. The rain outside didn’t stop them from using their trademark creativity to find ways to have fun indoors. We can’t wait for more discoveries on Day 3!
This year marks an exciting point in the history of CAMP: the program’s 10th anniversary! Starting today, the CAMPers (some returning, some new) are going to learn about combinatorics – the mathematics of counting and combining things.
First thing this morning, the CAMPers gathered under the staircase in the Reem-Kayden Center to try out a variety of Spot It! games – with themes like Pixar, Minions, Marvel, and Harry Potter. Each box has a deck of circular cards, and each card has a certain number of symbols on it. Any two cards will have exactly one symbol in common, no matter which cards you pick out. There are lots of ways to play, but most groups started with the simple yet highly competitive version: Whoever finds the common symbol first “slaps” the card and gets to take it.
After a quick orientation (and a “Happy Birthday” song for one of our CAMPers), the SINE group started talking about fractions – how to think about fractions as things (like apples or elephants) when considering operations like division of fractions. CAMPers learned that what we’re really doing when we divide 3/4 by 2/3 is put the fractions into a common denominator – 9/12 and 8/12 – then divide 9 pieces into groups of 8 pieces each. You would get one group with a remainder: 8/8 + 1/8 = 9/8.
“If I had 10 apples and I asked you to put 2 oranges into a group, could you do it? No, because apples aren’t oranges.”
“What we really need to do is let go of our predisposed notions and realize that apples ARE oranges.”
Next, they broke into groups to analyze the Spot It!decks: How many cards are there in each deck? How many symbols are there on each card?
Then Frances introduced them to a useful tool: Graphs. In the case of Spot It!, dots represent individual symbols and lines represent the connections between them – the cards they have in common. The CAMPers made graphs with pipe cleaner edges tied together at each node, while others chose to draw them on paper.
In Computer Science class, the SEC group started processing in Java – in other words, getting the computer to draw stuff. They learned about functions like size() (which controls the size of the canvas) and different shapes – for example, ellipse() . Given the function ellipse(100, 100, 50, 30); , the CAMPers were asked to “vote” on the following question: Where on the screen will the ellipse appear, and why? Next, they were given a coding challenge: Create a canvas of 800 by 800, then draw 4 ellipses, one at the center of each quadrant. Finally, they learned to use the rect() function to create – you guessed it – a rectangle and figured out the function to change the color of a shape.
After class, the CAMPers split up into groups, each with a fun elective to try: learning to juggle with Japheth, trying out pencil-and-paper puzzles like “Star Battle”, and learning how to solve a Rubik’s Cube with CS instructor Shiven. The CAMPers had a lot of fun with indoor activities, even though a thunderstorm was brewing outside.
After lunch, the CSC group had Computer Science class, where they learned about converting numbers to binary – the language of computers.
Meanwhile, the SEC group went to Art class to start brainstorming symbols for their own personalized Spot It! decks.
Finally, the CAMPers reconvened in the auditorium for a “math fair” of games, activities, and puzzles to choose from: paper puzzles, Rubik’s Cubes, Set, and of course, Spot It!
The first day of CAMP’s 10th year was full of fun puzzles, new challenges, and lots of laughs! Using their creativity to answer questions as they come up will set a great foundation for their explorations on Day 2.
It’s the fifth and final day of Bard Math CAMP. This morning opened with a special treat – the CAMPers finally got their Bard Math Circle t-shirts to wear! We got the Bitó Auditorium ready just in time for Leon Botstein, the president of Bard College, to pop in and say a few words about the importance of mathematics (and the importance of learning in general), as well as to answer a few by-the-numbers questions about Bard.
How many buildings are there at Bard?– Dr. Botstein says he hasn’t counted. However, this did lead into an interesting discussion about the amount of usable space in old buildings versus new buildings – estimating about 70% in RKC, with only about 40-50% in older buildings on campus.
How many students are there? – If we’re going to be counting students, we first have to define what a student is. Somebody who spends 100% of their time studying is a full-time student, whereas someone who spends about 50% of their time is only 1/2 student (two of those equals one full-time student). In Annandale-on-Hudson, Bard College has about 1,800 undergraduates and 200 graduate students; in NYC, Bard has about 80 grad students and 1,000 early-college students; in other U.S. cities, we have 2,500 students; the Bard Prison Initiative has 300 students; and throughout the rest of the world, about 2,000 students. This sums up to an estimate of approximately 7,880 students.
Why does Bard College have its own address? – Bard College comprises about 95% of the Village of Annandale, which is part of the Town of Red Hook. Annandale has its own zip code, 12504, and because of this it has its own federal mailroom (which is owned by the U.S. government). Since Bard College has its own zip code, it has its own address.
Bard College President Leon Botstein, holding a first-level Menger Sponge.
After that, the CAMPers headed to Math class – where COS worked on using the scaling factor (as well as various scientific calculators) to find the dimension of the Cantor Set. Next, they broke into groups to figure out the dimension of the Sierpinski Triangle and the Menger Sponge (a cube with a cube taken out from the middle, and so on with each of the smaller cubes of which it’s made).
Meanwhile, the SINE group used scientific calculators to find the area and perimeter of the Sierpinski Triangle at different stages. Then they used balls and sticks to build hypercubes (4-dimensional cubes), which they had heard about in the guest lecture on Wednesday.
After that, the SEC group made their way to CS class, where they finally found out the answer to a question that was probably always in the back of their minds: What really goes on inside a computer? Through a series of quests – Can you open the box? Can you find the “brain of the computer” (the CPU – Central Processing Unit)?How can you tell whether you’ve found it?What does the “motherboard” actually do?Can you find the Power Supply Unit (PSU), where the computer gets its power?Where is the Graphics Processing Unit? – CAMPers were introduced to the anatomy of a computer.
“Yeah, every computer has magic in it.” – Anish
After they finished their lunch at Kline, the CAMPers headed back to RKC for electives: continuing their speed-cubing workshop with Daniel Rose-Levine, learning to juggle, or helping to set up the photo booth for the upcoming afternoonopen house.
In Art class, SEC made finished up their artwork (while singing along to Hamilton!) to show to their parents later in the day.
Before the end of CAMP, students had a chance to showcase their work and to give their parents a taste of all the amazing Math, Computer Science, puzzles, games, and Art they had been learning (or creating) all week.
(A Tip for Parents:If you want to get an insider’s perspective of the math your child has been learning at CAMP, be sure to check their notebook!)
This year’s CAMP was a math-stravaganza, with lots of laughs along the way. Our CAMPers have gotten a chance to learn the kind of math you don’t have in school – the kind that shows how important it is to ask the right questions (not just find the right answer), try things out (even if you’re not sure they’ll work), work together, and remember to just have fun! Today, the CAMPers will bring home their t-shirts, artwork, and notebooks brimming with mathematical knowledge – as well as a creative learning mindset that will stay with them forever.
Photo Credit: Japheth Wood (image 4, featured image), Alexi Safford (images 5-8), Kateri Doran (images 1-3, 9-41)
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 Triangle – Why 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 tables – How 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).
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!
