We’ve just announced our amazing 2015 C.A.M.P. Staff! Please visit our staff page!
The CAMP is Waitlisted
The Creative & Analytical Math Program (CAMP) is currently full. To join the waiting list, please complete the application, as described on the CAMP Webpage.
CAMP is Waitlisted
The CAMP Program is currently full. We still welcome applications for the waiting list.
CAMP Registration is open!
The Bard Math Circle will offer its second annual Creative and Analytical Math Program (CAMP) for students entering grades 6-9 in the fall (priority will be given to rising 7th and 8th graders). Returning CAMPers are encouraged to apply!
Dates: August 24-28, 2015
Time: 9am – 3:30pm
Location: Bard College Campus
Cost: $180
Website: camp.bardmathcircle.org
Flyer: https://goo.gl/SnELdy
The program is intended for middle school students who have mastered the material in their math class, and are ready to tackle higher challenges. The curriculum includes mathematical problem solving, mathematical proof, computer science, and mathematical crafts. There is a focus on hands-on math, teamwork, insightful thinking and mathematical creativity.
The staff is composed of experienced math educators, many of whom majored in mathematics while at Bard College, and teaching assistants who are current undergraduate math majors. Our teaching goals are to expose local middle school students to high-level mathematical thought and topics not frequently encountered at school, and to inspire them to enjoy math and learn more.
To apply, please visit our CAMP website, CAMP.bardmathcircle.org. Please encourage your favorite middle school math student to apply!
Admissions is open for C.A.M.P. 2015!
We are pleased to announce that we will be holding our second summer of C.A.M.P.! The application is in the “How to Apply” tab and we encourage you to apply!
About:
The Bard Math Circle Summer C.A.M.P. (Creative and Analytic Math Program) is a week-long mathematics enrichment program that will expose students to challenging and exploratory mathematics. We welcome all students in Middle School to apply, however the program is primarily for students entering the 7th and 8th grades. Please take a look at our blog posts from last year to see what we learned at C.A.M.P.!
Date: August 24th – 28th 2015
Time: 9 AM – 3:30 PM
Tuition: $180 and Scholarships are available.
How to apply: Please visit our “How to Apply” tab for the application.
C.A.M.P. is over, for now!
Our first Bard Math Circle Summer C.A.M.P. was a success! We enjoyed a week of exciting mathematics with 23 middle school students the week before the school year, and you can read Justin’s reports on each day of the program in previous posts.
If you are interested in next summer’s program, please join our email list, below.
#mc_embed_signup{background:#fff; clear:left; font:14px Helvetica,Arial,sans-serif; } /* Add your own MailChimp form style overrides in your site stylesheet or in this style block. We recommend moving this block and the preceding CSS link to the HEAD of your HTML file. */
Subscribe to our mailing list
* indicates required
Email Address *
First Name
Last Name
Zip Code
Which best describes you? (e.g. student, parent)
I am interested in
- Library Programs
- Math Competitions
- Summer C.A.M.P.
//s3.amazonaws.com/downloads.mailchimp.com/js/mc-validate.js(function($) {window.fnames = new Array(); window.ftypes = new Array();fnames[0]=’EMAIL’;ftypes[0]=’email’;fnames[1]=’FNAME’;ftypes[1]=’text’;fnames[2]=’LNAME’;ftypes[2]=’text’;fnames[3]=’ZIP’;ftypes[3]=’zip’;fnames[5]=’ROLE’;ftypes[5]=’text’;}(jQuery));var $mcj = jQuery.noConflict(true);
Day 5
Today was our last day together! Although we had to part ways today, I’m sure that our mathematicians will continue their explorations within mathematics wherever they may go.
To help you on your way, here are some resources you might be interested in:
Books:
Websites:
What if? Answering “What if?” questions with math, physics, and more.
George Hart A mathematical sculptor, father of the famous ViHart.
Games:
Gomoku is also played on a Go board
Videos
Numberphile https://www.youtube.com/user/numberphile
MoMath Math Enounters https://www.youtube.com/user/MuseumOfMathematics/videos
Computerphile https://www.youtube.com/user/Computerphile
If you have anything that you would like me to add to this list, please email it to me at jjshin@optonline.net.
In our Proofs and Investigations module, we examined some hat problems that dealt with the issue of limited information. It reminded me of a similar, very famous problem about eye color. A version of it was written by Randall Munroe, I have transcribed it here:
A group of people with assorted eye colors live on an island. They are all perfect logicians — if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.
On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.
The Guru is allowed to speak once (let’s say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:
“I can see someone who has blue eyes.”
Who leaves the island, and on what night?
There is no easy trick to this question. No reflective surfaces or wording tricks. It is a solid, well known logic problem. Good luck!
We examined another infinite process today in Problem Solving, this one involving slicing and attaching squares. We derived a general formula for the perimeter using patterns we found in the length and the width of the object. We also attempted to find all of the ways four cubes can be combined to make unique figures. Thinking about this problem, I wondered how many different kinds of figures I could make with four tetrahedrons instead of four cubes.
Can you describe all of the polyhedra I can make using only four tetrahedrons that are attached face to face?
For our Mathematical Artifact of the day, we created hexaflexagons, figures that fold and unfurl to reveal patterns of shapes and colors. I used the cyclic diagram below to show all of the states and paths between every possible pattern on Siira’s flexagon.
 |
| Math |
Can you draw your own diagram that describes a flexagon that you made?
 |
| A mathematician shares his flexagon |
For our Puzzles and Logic module, we tackled a very famous probability paradox called the Monty Hall Dilemma. Faced with choosing doors hiding cars or goats, we developed a strategy that yielded success 2/3rds of the time! It is often used as a demonstration of how our intuition can lead us astray in probability problems like this one:
There are three boxes:
1. a box containing two gold coins,
2. a box containing two silver coins,
3. a box containing one gold coin and one silver coin.
After choosing a box at random and withdrawing one coin from that box at random, if that happens to be a gold coin, what is the probability that the other coin in the box is gold?
Be sure to justify your answer as we did for the Monty Hall problem!
For our Computer Science module, our mathematicians finished and shared the projects they have been developing. Some animations, interactive pictures, and games were made to show off our programming prowess.
 |
| Mathematicians share their programming work |
At the end of the day, we shared one last math salute as we parted for one last time. Our first Bard Math CAMP has come to a successful conclusion, and I am glad that I got to share some time with all of you.
Best Wishes,
Justin Shin
Day 4
Today was a day filled with shapes and colors of all sorts! In our Problem Solving module, we explored the volume and surface area of several Menger cubes. Using interlocking colored cubes to help us visualize the first step of creating a Menger sponge, we used patterns in the rise and fall of the volume and surface area to determine that a true Menger sponge has infinite Surface area but no volume!
 |
| Mathematicians discuss how to calculate the Surface Area and Volume of level one Menger cubes. |
Afterwards, we tried to build our very own Menger cube with some business cards one of our resident mathematicians found. Working together, we managed to construct some cubes of our own. We also got to continue our discussion on infinity we started yesterday in our Proofs and Investigations module.
 |
| Mathematicians begin forming the starting cubes for a Menger Sponge. |
A Menger cube is made by removing segments of a cube in an orderly fashion. Another fractal that works on a similar principle is a Sierpinski triangle. I have the first two steps shown below.
 |
| Image Credit: http://nrich.maths.org/4757 |
Can you describe what happens in each step? If the area of the first triangle is 1 unit squared, what is the area of the step 3 figure?
In our Computer Science module, we continued working with the Processing computer language. Learning about if-then statements and Boolean operations added more tools to use for our final project, which will be presented tomorrow. We all use “if ___ then”, “and”, “or”, and “not” in our daily conversations along with using them to help us code. My own Proofs teacher, Professor Ethan Bloch, proposed this question that tested my ability to understand logical operators:
If Susan likes fish, then she likes onions. If Susan does not like garlic, then she does not like onions. If she likes garlic, then she likes guavas. She likes fish or she likes cilantro. She does not like guavas. Can you tell if Susan like cilantro?
For our Puzzles and Logic segment today, we tried our hand at solving bridge-crossing problems. We used a game tree diagram to analyze the problem once we figured out some solutions. Good thing we went over a method of analyzing games, as I stumbled upon this unfinished tic-tac-toe puzzle:
It’s X’s turn. Can you draw a game tree describing all possible end states for this Tic-Tac-Toe board? How many different end states can this game result in?
 |
| A game tree for a river crossing problem. |
For our Proofs and Investigations module, we played around with one of my favorite environments for math problems, chessboards! As we attempted to tile chessboards with 2 x 1 domino pieces, I was reminded of a famous problem known to many mathematicians and chess players.
Can I place eight queens on a standard chessboard so that none of the queens are attacking one another?
When checking your solution, remember to check for diagonal attacks from each queen!
We have only one day left, and our resident mathematicians are busy preparing their last problem sets. Get ready, tomorrow we will be finishing up our final computer science projects as well as exploring some new problems in our modules.
-Justin Shin
Day 3
We are halfway through our time together, and we have built up our mathematical muscles with our Puzzles and Logic, Problem Solving, Computer Science, Proofs and Investigations, and Mathematical Artifacts modules. In fact, when I went to the doctor today, he told me that every day I go to Bard Math CAMP, the Math Lobe in my brain grows three times larger, and the Writing Lobe in my brain grows one and a half times larger. (I guess from writing all of these blog posts!)
If I started Bard Math CAMP with my Writing Lobe four times larger than my Math Lobe, on what day of camp will my Writing Lobe and Math Lobe be the same size? How many times larger will my Math Lobe be than my Writing Lobe at the end of the fifth (and last!) day of camp? How many times larger will my Math Lobe be than my Writing Lobe after n days?
The work we did today with ratios, and our work two days ago with Sequences might help with this one.
Our problem solving class worked on some good ratio problems today, first exploring fuel consumption on a car trip, and then examining a case of matching and mismatched socks. If you enjoyed our work with Ratios today, Frances (The lead mathematician in our Problem Solving module) gave me an extra problem to chew on, adapted from the New York Math League.
Mr. Zilch’s pocket just became empty. Nine minutes ago, his pocket was exactly half full. He was putting coins into it at a rate that would have filled his empty pocket in 36 minutes if it did not have a hole in it. If his pocket were full at the moment this hole occurred, and he didn’t add any coins, how many minutes would it take for the pocket to empty?
 |
| One of our mathematicians cooked up this handy diagram to help us think about fuel use on a car trip. |
We took another field trip today in Proofs and Investigations, but instead of traveling in a time machine, we all took a visit to Hilbert’s Hotel, the only infinite hotel in the world, to help Dr. Hilbert assign room numbers to his guests. Dr. Hilbert’s Hotel is a popular resting place for mathematicians, as they know that they can trust Hilbert to give them a room. But the infinite number of rooms isn’t the only thing strange about Hilbert’s Hotel, Hilbert’s Hotel also has a special Elevator. When you enter the elevator and press a floor number, for every minute that passes, the elevator will close half of the remaining distance. For instance, if I went into this elevator and pressed floor number 40, after the first minute it would take me to floor 20, and after another minute I would be taken to floor 30, and after another minute I would be taken to floor 35, and so on and so forth.
If a mathematician enters the elevator and presses floor number 128, how many minutes will have passed when he hits floor 124? The mathematician still hasn’t come out of the elevator, and it’s been about six hours… what’s taking him so long? Will he ever leave the elevator? Why or why not?
If you would like to learn more about infinity, here is an interesting video made ViHart, a mathematician youtuber.
For our computer science module, we learned about loops and how to use the setup function in the Processing computer language. Unfortunately, I wasn’t paying attention, and I need help interpreting a code that Yulia (Our lead mathematician in our Computer Science module) gave me.
int x;
int y;
void setup(){
size(500, 500);
x = 30;
y = 30;
}
void draw(){
background(0);
rect(x, y, 30, 30);
x = x + 1;
y = y + 1;
}
I can see that we are drawing a rectangle, but when I run this code, what direction will the rectangle move?
For out Mathematical Artifact of the day, we discovered made our own tessellation patterns!
Can you tell me how many different regular polygons can be tessellated on a flat surface? How do you know that you haven’t missed a tesselateable polygon?
Finally, for our Puzzle and Logic module, we tried to use probability to boost our odds at a guessing game. By analyzing the different possible combinations of red and blue dots, we could determine a strategy that gives us a correct guess about 75% of the time, greater than the 50% success rate of random guessing. If you liked this probability exercise, try this one, posed by Martin Gardener in his monthly column for Scientific American:
1. Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
2. Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
Are the probabilities the same?
… Are you sure?
Martin Gardner is famous for his column that explored a variety of fascinating mathematical puzzles. If you’d like some more, he has collected some of his most well known problems in this book.
As we pass the halfway point of our program, I can only look forward to what the next two days will bring. Together, we’ve already explored Graph Theory, Ratios, Path Counting, Loops, Modular Origami, Sequences, Variables, and many other facets of mathematics. Until tomorrow!
-Justin Shin
Day 2
Today in Puzzles and Logic, we shared a series of related mathematical games that involved sequentially taking coins from a pile. The “Don’t be greedy” game and the “Don’t be greedier” game are both known as games of “Nim”. Together, we discovered some strategies that are sure to succeed, as well as some winning and losing states. If you enjoyed playing the game of Nim, try this related problem given to me by my first competition math teacher:
Imagine you are playing a game against an opponent on a chessboard. A rook is placed on the bottom left square on the board, and both players attempt to move the rook to the upper right square on the board. The players take turns moving the rook. When a player moves the rook, they can move the rook any number of squares up or right, but they cannot move both up and right on the same turn. Whichever player places the rook on the upper right hand square wins.
-Mr. Holbrook
Does a winning strategy exist for the player that makes the first move? What about the player that makes the second move? What are some conditions that guarantee a win?
In our problem-solving unit, we discovered several different methods of deriving a general formula for the handshake problem proposed on the first day. We also started on a path counting problem, using the streets of New York City to think about the number of shortest paths between two points on a grid. Thinking about minimizing the amount of walking you need to do is very good for very tired people, and one of our mathematicians met a very tired ant that wanted to find the length of the shortest path between the circled corners on this 1m. x 1m. x 1m. wire cube.
 |
| Image Credit: Jürgen Kornmeier and Michael Bach |
Can you tell me the minimum distance the ant has to walk if the ant can only walk along the edges of the cube? How many “shortest paths” are there? How many “shortest paths” would there be if instead of a wire cube, we had a solid cube, and the ant could walk on the top of the faces of the cube instead of just on the edges?
Our computer scientists learned about variables today, and used them to help draw and color our own avatars on the Processing programming language. With our new tool, we figured out how to set or drawings to positions based on variables, so we could easily move our entire picture up, down, left, or right depending on how we set our variables in our code. Can you figure out how the variables in the following code will behave?
int cat = 18;
int dog = 36;
dog=dog/cat
cat=cat/5
dog=dog+dog+dog-cat
What is the value of dog?
During our Proofs and Investigations module, we took a trip in a time machine to the city of Königsberg, Prussia to explore Graph Theory as Leonhard Euler did in 1736. Using bridges and ghosts to help us think, we learned about Eulerian paths, which are paths on a graph that visit every edge once. If you enjoyed playing with Eulerian paths, you may want to take a look at Hamiltonian paths. A Hamiltonian Path is simply a path that visits every vertex exactly once.
Do you think I have to use every edge in a graph if I draw a Hamiltonian path? Do you think I can use an edge twice in a Hamiltonian Path? For a much tougher challenge, how many Hamiltonian paths can you find on a cube? I give you two examples of Hamiltonian paths on cubes below. The red line on the left cube makes the Hamiltonian path a Hamiltonian cycle, a special kind of Hamiltonian path that is a closed loop.
 |
| Image Credit: |
Finally, our mathematical artifact of the day was a triangle pattern that could be folded and glued to create an invertible three-dimensional figure. The mathematical artifact of today can be used to explore the path counting problems we talked about in our Problem Solving module, and the Eulerian paths we explored in Proofs and Investigations. Given two vertices, and assuming all edges are the same length, try figuring out how many shortest paths exist between different pairs of points on the figure. Alternatively, see if you can take a marker and draw out a Eulerian path using the vertices and edges of the figure as vertices and edges on a graph. Do these problems change now that we are using a distorted torus (A three dimensional solid that looks like a doughnut) instead of a flat plane?
Day 2 was packed with both tough problems and elegant solutions from our mathematicians. All of our mathematicians went home with lots to think about, and we await some more thinking and problem solving in the morning. Until tomorrow!
-Justin Shin