CAMP Staff 2015

2015 Staff:                    

Our enthusiastic and talented staff of math educators from Summer 2015!

Senior Staff 

Erin Toliver

After earning degrees in mathematics from Bard and Dartmouth Colleges, Erin became a math teacher at Saint Ann’s School in Brooklyn, NY. Now living in Toronto, she funnels her mathematical creativity into knitting, sewing, and crocheting projects.  She enjoys cooking, baking, running, and playing board games, but mostly finds herself chasing after her daughter who recently turned five-halves.

Frances Stern

Frances teaches math to teachers and students in New York City, working with struggling students and those eager for more and deeper math.  She has a master’s degree in mathematics and has written two books for parents and teachers, both titled Adding Math, Subtracting Tension, for grades pre-k to 2 and grades 3-5.  Drawing, painting, hiking and most recently folk dancing are favorite non-math activities.


Susan Tarnowicz

Susie is a visual artist who makes paintings, drawings, installations and short form writing. She graduated from the Rhode Island School of Design with a degree in Painting, and has since lived and worked in Italy, VT and NY as a dairy and vegetable farmer, participating in artist residencies, and teaching both middle school and high school art.  While teaching middle school girls she discovered the pragmatics of Mathematics in art education and designed curriculum to support learning in both subjects.   This summer Susie was awarded two artist residencies through the Byrdcliffe Colony in Woodstock NY and the The Dune Shacks in Cape Cod MA through Peaked Hill Land Trust for whom she feels extremely grateful.  Most recently Susie moved across the river to Germantown NY pursuing her career in Art Education, and is thrilled to be teaching Mathematical Art at the Bard Math CAMP.


Maureen Black

Maureen retired after teaching in Computer Science in high school for 33 years, and is now an adjunct professor at Marist College in Computer Science and Mathematics.  When not in the classroom, Maureen hosts open mics and performs with her husband in the folk duo Never Too Late.  Outside of math and computer science, her passion is for live music, most especially singer-songwriters performing their own original songs.

Junior Staff

Najee Mcfarland-Drye

Najee is a rising senior Math Major at Bard. He enjoys tutoring and volunteering for Bard Math events. His hobbies outside of math include bike maintenance, cycling, bicycle touring, and interacting with the amazing wildlife native to the Hudson Valley.








Zechen Zhang 

Zechen is a rising sophomore who intends to major in physic and math with a concentration in philosophy. He has participated in various math competitions throughout his education and during these was introduced to the joy of a challenging problem. When he isn’t holding paper and pen you can find him playing basketball, watching youtube videos and traveling anywhere in the world.

Jessica Liu

Jessica plans to moderate in math at Bard this upcoming year. She volunteered with the Bard Math Circle throughout the last school year and is very excited to engage with and inspire the students this summer. Outside of her mathematical interests she spends her time working on the Bard EMS squad and reading.

Andres Mejia 

Andres is a rising Sophomore at Bard College and is a native New Yorker from Queens.  He’s been independently studying advanced math through online course work since he was the age of the students coming to C.A.M.P.  Right now he’s excited about taking Real Analysis in the fall. In his free time he can be found playing guitar or listening to music.  


Program Director: Japheth Wood, PhD 

Japheth is a math professor at Bard College, whose involvement in the math circle movement has included directing the New York Math Circle, helping lead the Math Association of America‘s SIGMAA on Circles, as well as co-founding and co-directing the Bard Math Circle. Japheth envisions math circles as an effective way for mathematicians around the world to make a greater impact on math education at all levels, as well as opportunity to refresh and innovate their own teaching. He has also worked extensively with pre-service math teachers through Bard’s Master of Arts in Teaching program, supervised math research project in the Bard Prison Initiative, and is now working with the Bard undergraduate college. 

Program Director: Lauren Rose 

Lauren Rose has been a mathematics professor at Bard College since 1997.  She received her PhD in Mathematics from Cornell University and has taught mathematics at Cornell University, Rutgers University, Ohio State University, and Wellesley College, in addition to Bard College.  She is the co-founder of the Bard Math Circle and the Mid-Hudson Math Teachers Circle, which provide enrichment programs for middle school students and teachers in the Hudson Valley.  She is also Director of the Math Major in the Bard Prison Initiative.  Her current research interests include the Mathematics of Puzzles and Games, Voting Theory, and Integer Splines.

Program Coordinator: Eliana Miller

Eliana Miller is a senior at Bard College who has worked as the student coordinator of the Math Circle for the past two years. She is passionate about mathematical education outreach and is enthusiastic about the second season of C.A.M.P. this summer. Her academic interests are in logic and philosophy of math. She is currently working on her senior project on the combinatorics of KenKen Puzzles.


CAMP is over. For now!

Our 2015 summer C.A.M.P. (Creative and Analytical Math Program) is over, but our 2015–2016 program will be announced in the next few weeks. Please join our email list (Sign Up) if you are interested in our library programs, competition programs or C.A.M.P.

Day 5

Today was our last day of math CAMP! Fortunately, this didn’t mean that students didn’t continue to solve interesting problems.

We began our day by playing “Liar’s Bingo,” a game that utilized sequences of six different numbers that were chosen in order to follow a very mysterious rule. The process can be described as follows:

Each group of students sat together with some of these six number sequences, attempting to find any obvious patterns between them. The numbers were either contained in a red box, or a black box, suggesting some relevant relationship between color and the number contained. Since no student found any predictable pattern, we were able to safely have volunteers come to the front of the room to Eliana where the student needed read the sequence of colors on their strip, but also lie about precisely one of the colors in their sequence. from this information, Eliana was able to deduce the exact value of the number that was lied about.

An example strip from “Liar’s Bingo”

After successfully guessing many of the numbers, students then attempted to guess what the “trick” was behind Eliana’s odd ability to read their minds! One student in the back row suggested that the colors were actually being assigned values such as “0” for red, while another student suggested that perhaps the sequence of numbers could be subdivided into groups of three. Both of these suggestions lead to an entire class that had the answer on the tip of their tongue! However, the morning logic warm up concluded and students shuffled out of the room and into their other classes speaking to each other about what the trick might be.

Hint: Consider the fact that every number in liar’s bingo had either one or two digits. Now take a look at two distinct groupings of three numbers whose color seems to represent a particular value. Keep place value in mind! Hmm…

Here is a link where students can find their own “Liar’s Bingo” cards:

In geometry, Group A started with a hexagon template and were assigned to create their own design within the confines of the template. Here, the expectation was that students would use geometric principles in order to make shapes within the hexagon using only a compass and straight edge. Thankfully they had also been working on this technique in art class and were able to create many interesting designs.

Group B on the other hand used all of the  tools they had gathered during the week in order to construct a “Pi Sandwich.” Here, the goal was to understand how ancient civilizations were able to calculate Pi to an arbitrary number of decimal places! The stock answer for many middle schoolers in terms of how one might obtain the value of Pi is by dividing the circumference of a circle by its diameter. However, Group B should now understand that you can’t actually measure an irrational circumference to any degree of accuracy using this method. Instead, the group worked together with Frances in order to show that one can “sandwich” the circumference of a circle in between two hexagons (one that circumscribes the circle, and the other is inscribed.) This means that students were able to find lower and upper bounds for the exact value of Pi. Unfortunately, the Hexagon method seems to be insufficient as the bounds for Pi are still quite large with this method.

The first step of deriving Pi

With this understanding, students then worked in smaller groups to implement a similar method with a dodecagon (A shape whose perimeter is closer to the circumference of a circle.) While some groups didn’t come to an exact numerical estimation of Pi with the dodecagon,  many of them were able to see that one could use this method for an arbitrarily large “n-gon” and get the value of pi to as many decimal places as they had the patience for.

In graph theory, students were given an assortment of problems concerned with “tree graphs” and “bipartite graphs.” Unbeknownst to many students, these are actually very applicable forms of mathematical organization. A bipartite graph is a graph that can be partitioned into two different categories in which each “vertex” from one category is related to a respective vertex in the other. A natural formulation of a bipartite graph is to think of  a soccer game. Imagine one category that is the team to which each player belongs, and another category that consists of the players themselves. Draw an edge between a player and their team if and only if the player is in that team, and you will have a bipartite graph!

Here is another natural example of a bipartite graph:

Complete Bipartite graph
While trees and bipartite graphs are valuable tools to have, the techniques that students had to develop in dealing with these nuanced concepts might actually prove to be equally important. The act of actually proving some properties of trees and bipartite graphs (for example, that every tree is bipartite) is at once a difficult question and also a necessary one for any serious mathematician. For example, students learned about  “if and only if” statements, the basic premises of “induction,” and the use of definitions to form a “proof by contradiction.” Many students were alarmingly talented at finding counterexamples and using their own intuitions to decide on whether or not an idea was correct. The students worked in groups and were not only kind to each other but were able to actively articulate their thoughts so that everyone was able to understand how they reached their conclusion. In many ways, the last day of graph theory only continued to prove that children are far more capable than many adults would like to give them credit for! Congratulations to everybody who worked on some very good questions today.
Some additional problems to consider:
1) Explain why or why not every Eulerian bipartite graph has an even number of edges.
2) A Spanning Tree is a tree graph that connects every vertex of a connected graph G. Explain why every connected graph must contain a spanning tree.
3) Think of some real-life examples of a tree graph!
In mathematical art, all of this week’s work resulted in the completion of each student’s personalized three dimensional shape! Students on their final day began to seem like very independent artists, setting up their own drafting space and working steadily to complete their projects for this week. For students who were unable to finish, you all know the process now! See if you can finish your shapes at home and think about what materials you might need. 
Some of the artwork done by students
Some additional projects to consider include:
1) A small exercise in pythagorean color theory is the idea of how you can color the faces of each individual shape without any two adjacent faces containing the same color. The challenge here is to consider the minimum number of colors necessary to accomplish this goal.
2) A very important intersection between graph theory and the polyhedra we worked on this week is how each polyhedron can be drawn in a two dimensional graph! We can look at every corner of the polyhedron as a vertex, and every side of the shape as an edge that connects two vertices. Here is the graph for some familiar shapes, namely the hexahedron and tetrahedron:

As you guys can probable imagine, this can get very complicated very fast! Try doing the graphs for some of the other polyhedra we worked with this week. More importantly though, try to take your own shape and draw a graph of it.

**Challenge problem: Use some of the basic polyhedra that you know and draw their graphs. See if you can relate the number of faces, edges, and vertices in each graph.

In computer science, the students also completed their work with graphics and animation. As is apparent to anyone who saw their work, many of the students worked very intensively in order to create works of sophistication. More impressive is that students were able to learn an entirely new language and create something with it. Hopefully all of the projects have been shared with parents, and the students will use all of the language (as well as the downloadable software) they have learned about this week and continue to work on their craft.

One of the many computer science projects

All in all, it was a very successful day, and showed how valuable even a week can be in the development of any academic pursuit. I hope that all of the students continue to work independently but also strive to stay in touch with their like-minded peers. I also hope to see many of you back next year for another week of exploration and growth!

Good luck to everyone,

P.S If anyone has questions about any of these problems (or just want to talk about any of the math they’re doing during the academic year!) you guys can all reach me at
have fun with these interesting questions and all of the other great questions you can get your hands on this year.

A final goodbye, till next year!

Day 4

The mathematicians began their day as greedy pirates, swash-buckling, mathematically of course, to get the most gold coins in a famous game theory conundrum.

Here’s the problem for those who haven’t seen it before:

There are 5 rational pirates A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them.

The pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.
The pirate world’s rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates, including the proposer, then vote on whether to accept this distribution. In case of a tie vote the proposer has the casting vote. If the distribution is accepted, the coins are disbursed and the game ends. If not, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.
Pirates base their decisions on three factors. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins each receives. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal.The pirates do not trust each other, and will neither make nor honor any promises between pirates apart from a proposed distribution plan that gives a whole number of gold coins to each pirate.

Five pirates vying for 10 coins is an interesting problem but what if there were one hundred pirates each trying to get the most loot out of 100 coins? What if there were more pirates than coins? If there were more pirates than coins would the first pirate be able to get any coins? Is there a strategy where the first pirate would be able to stay onboard? Can you devise a formula for the number of coins any given pirate in such a situation would get where the number of pirates (N) is greater than the number of coins (G)? Use the backwards induction model like we used with the first version of the game!

Speaking of formulating your answer using variables, we continued to derive the pythagorean theorem in Geometry, replicating the work of the indian mathematician Bhakarsa in his famous proof. Walking the same path he did in 1114 CE!

In graph theory we built off of our definitions of days one through three to find isomorphic graphs  amidst a maelstrom of interconnected nodes and edges. Mathematicians then used this fresh skill set to draw their own planar graphs.

During lunch I noticed a young mathematician who was playing a game of Cat’s Cradle. I wondered, if they were to chop up the string could they make a planar graph with the same number of nodes and edges as we see in the picture below?

If you were to play cat’s cradle without chopping up the string, how many different non-planar graphs can you make? How many of these are identical? 

In computer science we talked about Boolean logic and the special definitions of and or or in programming.

Here’s a joke about the inclusive or:

A logician’s wife is in labor. The logician is in the waiting room.
The doctor comes out of the delivery room and says to the logician, “Congratulations, your wife gave birth to a beautiful baby!”
“Is it a boy or a girl?” asks the logician.
“Yes”, says the doctor.

In art, mathematicians continued to use their budding architectural acumen to make platonic solids. We’re eagerly shedding the templates that we made in the beginning of the week as tetrahedrons, octahedrons, dodecahedrons and more take form.

We’re disappointed that C.A.M.P. is discrete and not continuous. The last update (until next summer of course) is tomorrow, stay tuned!

— Eliana

Day 3

The warm up on Day 3 was slightly different than usual. Instead of doing one of the usual logic puzzles, the mathematicians had to figure out how to draw a perpendicular lines using a compass. Then they moved on to figuring out how to draw a pentagon using a compass, building on what they learned in art.

In Geometry, the young mathematicians saw the “Chinese Proof” of the Pythagorean Theorem, and figured out how the diagram proved it.

In Graph Theory, we defined what a complete graph is (a graph with an edge between every pair of vertices), and then the mathematicians found a formula for the number of edges in a complete graph with with n vertices.

We figured out that you can represent the number of edges in Kn with the series:

This doesn’t seem like the most elegant formula however. Can you think of a way to write the formula that doesn’t need to use that pesky ellipses?

Then we used the Pigeon Hole Principle to prove that every graph has at least two vertices of the same degree. The Pigeon Hole Principle states that if you have pigeons, and n-1 pigeon holes, then at least one pigeon hole will have two or more pigeons.
For those that were wondering, pigeon holes look like this:

Can you think of any other scenarios where the Pigeon Hole Principle would apply?

During activities we split off in to groups, and some of us went to the waterfall!

To keep safe, we used the buddy system to keep an eye on each other. Since there was seven of us, there were two pairs of two buddies and one group of three buddies. How would you represent this with a graph? Would it be a connected graph? Would it be a simple graph?

In art the mathematicians continued the work they did yesterday and started cutting and assembling their polyhedron outlines.

In computer science the two groups embarked on different missions. Group A continued coding the drawings they designed themselves. The drawings ranged from bears to pizza to emojis to robots, all very cool things. Group B started coding their own computer game, which when completed will involve trying to catch a dropping ball with a block on the bottom of the screen.

That’s all for today. Check back again for more exciting mathematical adventures!


Day 2

We began our day with a logic puzzle that goes by the moniker, “Impossible Elevators.” Although, the bright mathematicians got to work making it possible to get to any floor in a seven floor building with six elevators granted each elevator could only stop at three floors and you’re not allowed to switch elevators! Can you figure out how?

What if each elevator could stop at 6 floors, you still have six elevators, and you can still get to any floor from any other floor granted you find the right elevator — how many floors could this building have?  Can you use your answer from the first part to guide you?

As we’ve seen in our Geometry Module oftentimes numerical patterns have geometric representations. In this case, the nifty Projective Plane of Order 2 also known as a “Fano Plane” may help you muddle through this elevator enigma.

Speaking of geometric representations, in Geometry mathematicians transformed into architects to make tessellations out of the triangles we made on Day 1. These tessellations led to new revelations. We wrote proofs for the triangle inequality and created a derivation of the area of a triangle; centuries of mathematical research in just one day!

On the topic of math research — Graph Theory took us to Leonard Eulors theorem for the Eulerian Path, and our young mathematicians began constructing their own algorithm’s for a Eulerian path.

Is there a Eulerian path in the Projective Plane of Order 2 shown above? Can you write your own algorithm for traversing any Eulerian path? 

In art we continued to use the tool of the week, the compass, to construct elaborate shapes. Today we  drew tetrahedrons, also known as pyramids.

Using your knowledge of drawing triangles with a compass can you figure out how to construct an octahedron, a dodecahedron or an Icosahedron


 Mathematicians made nets (outlines) to later create polyhedrons just like the ones above!

Lastly, I promised to share with you the answer to the pesky Egg Drop Problem, if you haven’t solved it yet — steer clear of what’s next.

Here is the complete problem for reference. Take a moment to solve it if you haven’t already:

You are assigned the task of determining from which windows in a 36-story building it is safe to drop eggs from. We make a few assumptions:

  • An egg that survives a fall can be used again
  • A broken egg must be discarded
  • The effect of a fall is the same for all eggs
  • If an egg breaks when dropped, then it would break if dropped from a higher window
  • If an egg survives a fall, then it would survive a shorter fall

You only have two eggs with which to test the drops. What is the least number of egg-droppings that is guaranteed to work in all cases?

Spoiler Alert! Answers Below

I talked to a few mathematicians at C.A.M.P. who cleverly figured out that they didn’t have to drop the egg from every single floor to know exactly where it would break. They could split the floors in half. They would drop the first egg from the 18th floor, and then if it broke they would start dropping the second egg on the 1st floor, dropping each floor up until the 17th floor. If the egg survived they would drop it on the 36th floor. If it broke there they would repeat the same strategy with the second egg, except start on the 19th floor and drop all the way up to the 37th floor. They would know exactly where the egg broke while saving themselves 18 unnecessary drops!  Not bad, using this strategy, in the worst case scenario they could accomplish the task in 19 drops.

Some mathematicians went even further, they figured they could split the floors into thirds and drop the egg every 12 floors. Using the same strategy as with every 18th floor, they would drop the first egg a maximum of three times (once on the 12th floor, once on the 24th floor and once on the 36th floor) and then they would drop the second egg a maximum of 11 times (up from the last interval where it survived to the interval where it broke).

What if we split the floors up into intervals of 4, 5, 6, 9, 12 and 18? Using the same strategy as above and imagining the worst case scenario, we get the following chart to describe the maximum number of egg drops for each interval.

Floors Skipped In between Drops
Maximum # of First Egg Drops
Maximum # of Second Egg Drops
Maximum Number of Total Egg Drops

What do you notice? It seems that two numbers are important. The maximum number of times you can drop the first egg and the maximum number of times you can drop the second egg. It appears that the closer those two numbers are together, the smaller the total number of drops is.

So, while intervals of 6 (with 6 and 5 as the two numbers) is a great answer — giving you a maximum of only eleven drops! It would be better if the two numbers are exactly the same!

But how can we do that? 

What if we didn’t divide the intervals equally? 

Is there a series where the number in the series is equal to the difference between that number and the last? There is, the triangular numbers have this pattern!

What triangular number series adds up to 36? 


8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36

How would a series of drops that travels along the triangular series of 8 look? You would drop from the 8th floor then hop up 7 floors to the 15th floor then hop up 6 floors to the 21st floor, ect.

If you get up the 8th floor and it breaks then you have to try floors 1-7 giving you a total of eight drops.

If you get up to the 15th floor and it breaks, you’ve already used two drops and then you have to try floors 7-15 giving you a total of 8 drops.

If you get up to the 21st floor and it breaks, you’ve already used three drops and then you have to try floors 15-21 giving you a total of 8 drops.

So the maximum number is always 8 drops! That’s the least that we can possibly do it with, since the number of intervals traversed is always the same as the number in-between the intervals

That’s all for now! More to come soon!

— Eliana

Day 1

The day began with our young mathematicians looking at a puzzle about dropping eggs from tall buildings. Here’s a variation of the problem that you can mull over. 

You are put in charge of testing the durability of a new cellphone. You have two cellphones and a 100 story building. You want to know at which floor the cellphone will break. If the cellphone breaks when dropped from a a certain floor it will also break at any floor higher. If a cellphone survives a drop it will survive if dropped from a lower floor. How can you minimize the amount of times you have to drop the cellphone? 

Can you come up with a scheme for dropping the cellphones so you don’t have to stop at every floor? How would you skip floors? We’ll be revealing the answer in a later post, so stay tuned. Hint: It involves magical triangular numbers! 

Speaking of triangles and their mystical properties, in our geometry class we looked at sets of three natural numbers that add up to 18. Then we dusted off our compasses and got to work constructing triangles with edges of the lengths of these triples. A strange property emerged before our mathematicians own eyes! 

Triangle architecture aside, we ventured to a town with the problem of not knowing which bridges to construct. I’m talking of course about the famous Seven Bridges of Königsberg Problem. Our mathematicians travelled the same path as Euler in the early 18th century down the bridges of Konisberg to see if they could construct a route through the city crossing each bridge once and only once. The thinkers dipped their toes into the world of graph theory. 

Just like the bridges in a city in Prussia, what else can we use graph theory to represent? How about the paths that mathematicians could take to get to the Reem-Kayden Center each morning for C.A.M.P. Let’s say you were to park in the Reem-Kayden Center Parking lot and eventually make your way to the Reem-Kayden Center Building, but you want to cross through every path on central campus once and only once. Can you do this? 

Does the parity of the nodes and edges matter? Which areas of campus can you traverse using this method? 

In computer science we dabbled in a different sort of pairwise relationship — the relationship between  the command you would give a computer program and the output. Mathematicians coded for shapes to see that making an ellipse isn’t so different from making a circle once they figured out the geometric difference between them. 

Can we use those same geometric properties to make artwork? In our last module of the day, the thinkers took our their trusty compasses again to make tessellation’s and fractals out of a variety of polygons and circles. In doing so they used properties like equidistance and symmetry. 
Walking around the class, I happened to speak to a mathematician whose art looked like this: 

Which poses an interesting question! If each circle in this drawing contains a set of triples that adds up to 18, can we fill it in without writing numbers twice? How would the triples from Geometry look in this artwork? Is there any aesthetically appealing symmetry? 

That’s all for now, keep posted for more updates throughout a week of mathematical explorations. 

 — Eliana