Author: Bard Math Circle

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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:
Martin Gardener Books 
Shapes Space and Symmetry 
The Code Book 
The Simpsons and Their Mathematical Secrets 
How to Lie with Statistics
Great Book on Game Theory
Websites:
What if? Answering “What if?” questions with math, physics, and more. 
WolframAlpha 
George Hart A mathematical sculptor, father of the famous ViHart.
Games:
Ancient Greek Geometry
Go Board Game 
Gomoku is also played on a Go board
Videos
Vihart https://www.youtube.com/user/Vihart
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
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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
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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
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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: mathafou.free.fr

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
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Day 1

Things got off to a great start today, as our mathematicians started with a coin division problem that led to some interesting revelations about strategy and rationality. The problem, famously known as the Pirate game, asks how five pirates can distribute a treasure chest of gold amongst themselves using a set voting system.
If you liked the Pirate game, you might also want to consider another famous game in the discipline of Game Theory.
An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical items. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of $100 per suitcase (he is unable to find out directly the price of the items), and in order to determine an honest appraised value of the antiques the manager separates both travelers so they can’t confer, and asks them to write down the amount of their value at no less than $2 and no larger than $100. He also tells them that if both write down the same number, he will treat that number as the true dollar value of both suitcases and reimburse both travelers that amount. However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: $2 extra will be paid to the traveler who wrote down the lower value and a $2 deduction will be taken from the person who wrote down the higher amount. The challenge is: what strategy should both travelers follow to decide the value they should write down?
-Kaushik Basu
If you were a traveler faced with this situation, what value would you submit to the airline? Would this value change if you were playing this game against a perfectly rational computer?
While we were checking all of our mathematicians in, everyone received a t-shirt with a spiral pattern drawn on it. Luckily, our Proofs and Investigations module did some sequence and series work that might shed some light on the nature of the spiral pattern. How do you think that spiral was constructed? This spiral is often referred to as a “Fibonacci Spiral”, after the Fibonacci sequence which goes 1, 1, 2, 3, 5, 8, 13… how do you think the spiral and sequence are related?
In our Problem Solving module, we introduced a handshake problem to get to know one another while doing some good math. The problem we proposed asked how many handshakes would occur if everyone in the room shook hands with everyone else exactly once. Unfortunately for me, I got a little confused when I tried to duplicate the problem at a riddling party I attended. Instead of calculating the number of handshakes from a given number of people, I tried to find the number of people given the number of handshakes. In the riddling party, everyone shook hands with everyone else exactly once, and there were 4,656 handshakes that occurred. Can you help me find out the number of people that were in the party?
The Handshake Problem: Students found many ways to approach the discovery of a general rule.

Stopping by our Computer Science Module, I saw our mathematicians exploring how to give instructions to computers in the form of code. I’m glad that I came into the class, because I have been having a lot of trouble trying to code my personal robot to go about doing my personal routine. If you had a robot that looked exactly like you, how would you instruct it to go about your daily morning routine? For example, putting on a pair of pants might go something like:

1.  Locate pair of pants.
2. Orient pair of pants so that the zipper is facing away from robot and leg holes are pointing down. 
3. Place the right leg into the rightmost leg hole.
4. Place the left leg into the leftmost leg hole.
5. Shimmy pants up so that the belt loops are at the waist level.
6. Zip up and button pants.

Something that might be very simple to you might be pretty complicated for a robot. You may end up with a large number of instructions just to make sure your robot doesn’t go out the door with it’s pants backwards or on top of it’s head.
All in all, we had some great breakthroughs working with our mathematicians. I’ll leave you with one last math moment, from our mathematical artifacts module. We made figures out of Sonobe units today, and as I stacked three Sonobe unit cubes on top of one another, I wondered how many Sonobe units it would take to build a hollow rectangular prism with the same dimensions as my three stacked cubes.
Stay tuned for more blog posts on our progress as the week goes on! I’m sure we’ll find some more hidden problems to share with you all on this blog.

-Justin Shin
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Bard Math Circle Day 0

It’s the day before Bard Math CAMP, and our resident mathematicians have been busy getting ready for arrival day. The 0’s need to be properly inflated, our curly bracket’s (For our Computer Science minions) must be gone over with a curling iron, and our Greek letter’s must be taken out, sorted, and thoroughly dusted. Despite the work, everyone prepping for Bard Math CAMP has had some time to do their daily regimen of mathematical exercise.

In fact one of our mathematicians noticed something interesting about the number of campers arriving tomorrow. While we were celebrating the birthday of Bartholomeo Pitiscus, the mathematician who coined the word “Trigonometry”, a resident mathematician decided to calculate the probability of at least 2 of our 23 campers sharing the same birthday.  Try the problem out for yourself! Is the result what you had expected?

Another of our mathematicians made this logic puzzle after observing the areas where our mathematicians come from.

We have 23 mathematicians coming from Arlington, Bethlehem, Brewster, Catskill, Hyde Park, Kingston, New York City, Onteora, Red Hook, and Rhinebeck. Six areas have exactly one mathematician. Rhinebeck has one quarter the number of mathematicians that Kingston has, and Kingston has four times the number of mathematicians that Onteora has. There are exactly two areas that have two mathematicians. Neither Rhinebeck nor Red Hook have only one mathematician. From this information, can you find out how many mathematicians are coming from each area?

One last math moment to share!

I noticed that if I represent the number of eighth graders as “n”, the number of seventh graders is 2n-1, the number of boys is 2n+1, and the number of girls is n-2. Given that we have 23 incoming mathematicians that are either in seventh or eighth grade, and are either a boy or a girl, can you figure out how many mathematicians are in seventh grade? Eighth grade? Are boys? Are girls?

This is only a taste of what’s to come! As we get ready to welcome new mathematicians to Bard, we are sure to stumble upon some more problems hiding under the floorboards, behind whiteboards, and all sorts of unexpected places. Hope to see you soon!

-Justin Shin

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Fall 2014 Programs

Congratulations on the start of the school year! Bard College and the surrounding school districts are coming back into session, and we’re excited about math circle programming for the fall.

Events at Bard College, Fall 2014

Saturday, October 18, 1-3pm. Math Circle, including puzzles and games, challenging problems, and a hands-on project that students can take home. We help students strengthen their critical thinking skills and make math more fun.

Saturday, November 15, 1-3pm. Math Circle, including puzzles and games, challenging problems, and a hands-on project that students can take home. We help students strengthen their critical thinking skills and make math more fun.

Tuesday, November 18, 4-7pm. The AMC 8. This Contest contains engaging math problems that are challenging at the middle school level. The exam is intended to inspire, promote enthusiasm, and a healthy attitude towards mathematics. Students will be exposed to the richness of middle school level mathematics at a deeper level than ordinarily encountered in the schools. And after the exam, students will be treated to an engaging math talk from a Bard math professor.

Each of these events is free, but students must register in advance. A registration form will be posted on this page in September. If you’d like us to email you the announcement for when registration opens, please send us a friendly email to bardmathcircle@gmail.com.

Library Programs

Each month also features a math circle at the Kingston and Tivoli libraries. Visit our library page to learn more!

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Euclid the Game

A really nice website just stumbled upon is Euclid the Game.

This game starts by challenging you to construct an equilateral triangle, given a segment, with just 3 simple tools that approximate a straightedge and compass.

From there, you progress through some standard constructions that everyone should know (but none do, of course – that makes it more fun). I can’t wait to see how far these constructions progress.

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Bard Math Circle Survey – please participate!

Dear Math Circle Participant,
Please help us measure the impact of our math circles by participating in the following math circle survey:

Math Circle Survey: http://goo.gl/vO77DM

This survey is being conducted by the National Association of Math Circles (NAMC), a program of the Math Science Research Institute (MSRI) of Berkeley, CA. The Bard Math Circle has received support from NAMC in the form of several mini-grants over the years, which has helped us provide math enrichment programming at no charge. For more details, you can read our annual reports at the bottom of our documents page: http://www.bardmathcircle.org/p/math-circle-problem-sets.html
The math circle movement in the US is at a very early stage of development, and mathematicians across the country are sharing math problems, lesson plans, and developing best practices. NAMC and other groups support math circle activities, and seek funding through grants and donations. This survey is intended to quantitatively measure the impact of math circle activities, and this information will be of enormous help in improving programming, and gaining additional funding.
To participate in the survey, your parent or guardian must sign the waiver form on the first page of the survey. It’s a little over the top in terms of legalese, but that is standard for high quality surveys designed to collect data for educational studies. Please do participate!
Sincerely,

Japheth
—

Japheth Wood, PhD
Bard Math Circle