Albany Area Math Circle will be starting up its 13th year!

Our high school group will start meeting soon. We welcome new students in grades 9 to 12 to join us. To stay informed with all the information you need about place/day/time, please sign up for our email list by sending an email to AlbanyAreaMathCircle-subscribe@yahoogroups.com.

Students in middle school (grades 8 and below), we have not forgotten about you. Our high school students will be making plans for fun events to welcome you into our mathematical community later this year. Your parents are also welcome to sign up for our email list (using the same address as above) so your families can stay informed about those plans.

## Sunday, October 20, 2013

## Wednesday, May 29, 2013

### Recommended summer reading for young students (and their parents!) aspiring to climb mathematical mountains together this summer

Imagine if one of your daughter's classmates had an MIT professor dad who loved the fun of mathematical problem solving in his spare time. Dream on and imagine that he volunteered to share his enthusiasm and talents as a mentor with a small group of students including your child, busting them out of the conventional curriculum hamster wheel to take them on challenging mathematical rock-climbing adventures with inspiring views of beautiful mathematical mountain vistas.

Glenn Ellison's daughters are fortunate to have just such a dad and this engaging book is the result of his very successful mathematical excursions with his daughters and their schoolmates. Some of the students with whom he has worked for a number of years have now grown into world-class problem solvers.

Written in a good-natured conversational style,

*Hard Math for Elementary School*lays the foundation for elementary school students to develop the tools and habits of confident, capable, and curious problem solvers. The text provides well-organized explanations and the accompanying workbook poses thoughtfully composed practice problems designed to inspire children to tackle tough problems that exceed the expectations of conventional textbooks. This book and its earlier counterpart for somewhat older students, Hard Math for Middle School, are great solutions to questions frequently posed by parents of young students looking for summer reading for their mathematically voracious students.

*Groweth sed and bloweth med*

*And springth the wude nu,*

*Sing cuccu!*

(Disclosure: thanks to Professor Ellison for sharing a prepublication review copy of the manuscript with me.)

## Thursday, April 4, 2013

### Doing justice to describing the work of other math circles that have inspired us

Last week, Sol Lederman interviewed Gili Rusak and myself for a podcast now featured on his blog, Wild About Math. Thanks to Catherine Miller for typing up a transcript of that conversation, which will soon be available in a link on Sol's blogpost. Reading that transcript of an informal live unrehearsed and unscripted conversation was certainly humbling and subsequent reflection made me realize that there are things I need to clarify. There were so many things that I wish I had said, inspiring people who have massively contributed to our math circle I wish I had named or things I had said a bit more clearly. In one case, what came out of my mouth (when I was talking about "ends and means") was totally the opposite of what I intended to say and changed the meaning entirely. So I have added annotation in brackets to the transcript, and I have also provided links.

It was a fun experience talking to Sol, who is clearly a kindred spirit, an amateur math-lover like the two of us who shares our passion for promoting math communities where people enjoy celebrating mistakes and sharing Aha! experiences as they explore challenging problems together. We touched on many subjects and definitely did not have time to do justice to all of them in an hour-long informal conversation.

I want to acknowledge here in this blog an important distinction which we did not make in the podcast and which I have also neglected to make in the past in this blog, and which Ken Fan, the mathematician who directs Girls' Angle, and invented the treasure hunt concept, has called to our attention and asked that we clarify. There is a good deal of difference between SUMiT, the original treasure hunt created by Girls' Angle and the small local treasure hunts in Schenectady inspired by it. We have not described the extremely rich complexity of the far more elaborate original SUMiT event Gili attended. Although Ken has asked that SUMiT participants not disclose the full details of that experience (in order not to spoil the story line for future participants), he would like to clarify that the original event is far more complex with several stages, and the crossword element described by Gili in the podcast was only one of those stages. Ken has told me that thousands of hours of work have gone into creating and developing the SUMiT event. Prizes given to all participants included stereo speakers, a backpack, a set of Zometools, a Tetraxis puzzle from KO Sticks, candy, and a copy of Maria Dzielska's book

In the podcast, we also talked about some of the other mathematical circle communities which have inspired us and which are run by full-time professional mathematicians who dedicate their lives to creating mathematics and mathematical communities, but again I feel I did not do them adequate justice in giving them the credit they deserve for the inspiration they have provided to our math circle. I am in awe of them and have long and fervently wished that our own local community had such dedicated fulltime professional mathematicians leading a local math circle as Harold Reiter from the Charlotte Math Club, Bob and Ellen Kaplan from the Boston Math Circle, Zvezda Stankova from the Berkeley Math Circle, Tatiana Shubin from the San Jose Math Circle, Paul Zeitz and Brandy Weigers from the San Francisco Math Circle, Joshua Zucker from the MSRI Julia Robinson Math Festivals, Amanda Serenevy from the Riverbend Community Math Center, Ken Fan from Girls' Angle, or Japheth Wood from the NYC Math Circle and Bard Math Circle.

I also wish that I had encountered such people and communities when I was a student myself. I missed out on a lot of joy as a result when I was young. I never dreamed that math would be fun to do as a student or that it would be fun to do with other people rather than as a solitary pursuit. Indeed, when I was Gili's age, math was my weakest subject and I remember feeling quite lost and confused in my 10th grade geometry class (a much less advanced math class than she is taking. Like a number of our math circle students, Gili is already taking advanced college math courses.) It was only later when my younger brother and sister--out of desperation and having nobody else to turn to as they were entering a new high school where they did not want to be behind the other students--started asking for my help in learning mathematics that I began to get an inkling that it could actually be fun to figure out how to work together with them to try to find answers for their questions and to share my own (generally half-baked) insights. In some sense, we are still doing that in our math circle all these years later.

I also want to make clear that--in default of such folks living in our midst--our Albany Area Math Circle activities are all led by adult and student volunteers, amateurs who do not have the mathematical sophistication of full-time professional mathematicians who have spent their lives immersed in mathematics. While we have benefited from the countless wonderful ideas generously shared by professional mathematician math circle leaders and especially the facilitation of MSRI National Association of Math Circles in providing opportunities for them to share their ideas with us, we have no illusions that the depth of our mathematical understanding of all the nuances of the problems we investigate is anything like theirs or that we are able to describe them fully, but we do encourage others not to give up altogether if they also find themselves in a community lacking such dedicated professionals.

I encourage all our members as well as readers of this blog thinking about starting their own local math communities to go directly to the source--and also to support the math circles led by professionals whose work you admire by considering purchasing some of the MSRI Math Circle book series, Bob and Ellen Kaplan's books, subscribing to the

It was a fun experience talking to Sol, who is clearly a kindred spirit, an amateur math-lover like the two of us who shares our passion for promoting math communities where people enjoy celebrating mistakes and sharing Aha! experiences as they explore challenging problems together. We touched on many subjects and definitely did not have time to do justice to all of them in an hour-long informal conversation.

I want to acknowledge here in this blog an important distinction which we did not make in the podcast and which I have also neglected to make in the past in this blog, and which Ken Fan, the mathematician who directs Girls' Angle, and invented the treasure hunt concept, has called to our attention and asked that we clarify. There is a good deal of difference between SUMiT, the original treasure hunt created by Girls' Angle and the small local treasure hunts in Schenectady inspired by it. We have not described the extremely rich complexity of the far more elaborate original SUMiT event Gili attended. Although Ken has asked that SUMiT participants not disclose the full details of that experience (in order not to spoil the story line for future participants), he would like to clarify that the original event is far more complex with several stages, and the crossword element described by Gili in the podcast was only one of those stages. Ken has told me that thousands of hours of work have gone into creating and developing the SUMiT event. Prizes given to all participants included stereo speakers, a backpack, a set of Zometools, a Tetraxis puzzle from KO Sticks, candy, and a copy of Maria Dzielska's book

*Hypatia of Alexandra*.In the podcast, we also talked about some of the other mathematical circle communities which have inspired us and which are run by full-time professional mathematicians who dedicate their lives to creating mathematics and mathematical communities, but again I feel I did not do them adequate justice in giving them the credit they deserve for the inspiration they have provided to our math circle. I am in awe of them and have long and fervently wished that our own local community had such dedicated fulltime professional mathematicians leading a local math circle as Harold Reiter from the Charlotte Math Club, Bob and Ellen Kaplan from the Boston Math Circle, Zvezda Stankova from the Berkeley Math Circle, Tatiana Shubin from the San Jose Math Circle, Paul Zeitz and Brandy Weigers from the San Francisco Math Circle, Joshua Zucker from the MSRI Julia Robinson Math Festivals, Amanda Serenevy from the Riverbend Community Math Center, Ken Fan from Girls' Angle, or Japheth Wood from the NYC Math Circle and Bard Math Circle.

I also wish that I had encountered such people and communities when I was a student myself. I missed out on a lot of joy as a result when I was young. I never dreamed that math would be fun to do as a student or that it would be fun to do with other people rather than as a solitary pursuit. Indeed, when I was Gili's age, math was my weakest subject and I remember feeling quite lost and confused in my 10th grade geometry class (a much less advanced math class than she is taking. Like a number of our math circle students, Gili is already taking advanced college math courses.) It was only later when my younger brother and sister--out of desperation and having nobody else to turn to as they were entering a new high school where they did not want to be behind the other students--started asking for my help in learning mathematics that I began to get an inkling that it could actually be fun to figure out how to work together with them to try to find answers for their questions and to share my own (generally half-baked) insights. In some sense, we are still doing that in our math circle all these years later.

I also want to make clear that--in default of such folks living in our midst--our Albany Area Math Circle activities are all led by adult and student volunteers, amateurs who do not have the mathematical sophistication of full-time professional mathematicians who have spent their lives immersed in mathematics. While we have benefited from the countless wonderful ideas generously shared by professional mathematician math circle leaders and especially the facilitation of MSRI National Association of Math Circles in providing opportunities for them to share their ideas with us, we have no illusions that the depth of our mathematical understanding of all the nuances of the problems we investigate is anything like theirs or that we are able to describe them fully, but we do encourage others not to give up altogether if they also find themselves in a community lacking such dedicated professionals.

I encourage all our members as well as readers of this blog thinking about starting their own local math communities to go directly to the source--and also to support the math circles led by professionals whose work you admire by considering purchasing some of the MSRI Math Circle book series, Bob and Ellen Kaplan's books, subscribing to the

*Girls' Angle Bulletin,*participating in a future SUMiT, and/or just making a donation directly to any of these circles that has provided ideas you have especially enjoyed. These worthy organizations typically operate on very fragile financing and deserve your support. The end of the school year will soon be upon us, and I know that many math teachers would be delighted to receive such a book, a subscription, and/or a thoughtful donation made in their honor rather than the usual end-of-year thank you teacher gifts like candy or toiletries. It's a gift that will keep on giving for many years to come as your teachers' future students will benefit in many ways. You can also help support these generous professional math circle leaders by suggesting to your friendly local librarian that the library consider purchasing or subscribing to their publications.## Wednesday, March 27, 2013

### Math Circle student research honors

Math Circle students Gili Rusak, Matthew Babbitt, and Zubin Mukerjee have won honors this year for their original math research projects. They will be presenting their research in separate sessions at the 20th annual Hudson Undergraduate Math Research Conference, which will be held at Williams College on Saturday April 6.

Gili's applied mathematics project,

Matt Babbitt's graph theory research project,

Zubin Mukerjee's number theory research project,

Gili's applied mathematics project,

*An Analysis of Teenage Twitter Communities*, which draws on graph theory, probability, and sociology, won top honors at Capital Regional Science and Engineering Fair at RPI last week. This means that Gili will represent our region at the Intel International Science and Engineering Fair (ISEF) to be held in Phoenix, Arizona. This is the world's largest international science fair, where over 1,500 students from over 70 countries around the world gather to present their work. Over $3 million in prizes will be awarded. Gili is a sophomore at Shaker High School who has been taking advanced college classes at Siena College. You can read more about Gili's awesome work as a mathematical community builder here.Matt Babbitt's graph theory research project,

*Counting number of edges, thickness, and chromatic number of k-visibility graphs,*won semifinalist honors in this year's Intel Science Talent Search. Matt's research benefits from advice from his MIT Research Science Institute mentor, Jesse Geneson, and Dr. Tanya Khovanova, the head mathematics mentor at RSI. Matt, a homeschooled senior from Fort Edward who has taken advanced math classes at Union College, has been named a Jack Kent Cooke Scholar and plans to attend MIT next year.Zubin Mukerjee's number theory research project,

*Random Involutions and the Number of Prime Factors*, is based on his joint work with a fellow student, Uthsav Chitra, at PROMYS last summer. Their research mentor was Dr. Kristen Wickelgren, a research fellow at Harvard. The project won semifinalist honors in the*Siemens Competition in Math Science and Technology*. Zubin, a senior at Guilderland High School, who has been taking advanced math classes at SUNY Albany, has also won a number of honors in history and music. You can read more about Zubin here.## Friday, February 8, 2013

### Gili Rusak, mathematical community builder

Gili Rusak launches students on investigations of Archimedean solids. |

Gili Rusak, a tenth grader at Shaker High School who also takes advanced math classes at Siena College, has been building deep, rich, and inclusive mathematical communities all around the Capital District and even beyond. For the past two years, she has been helping Doyle Middle School teacher Nancy Smith with coaching Doyle's MATHCOUNTS team in Troy.

Last winter, she participated in the first annual SUMiT, a "fully collaborative, math intensive event" organized by Girls' Angle and the Undergraduate Society for Women in Mathematics at MIT and returned home efferverscent with enthusiasm about the wonderful experiences she had had as a participant in that event. She came back inspired with a missionary zeal to create a similar math event here in the Capital District, to bring that same mathematical joyful collaboration to students in the Capital District.

After months of thoughtful planning and brainstorming, Gili designed, organized, and led a completely marvelous math treasure hunt inspired by the SUMiT model. Gili's local event took place at Union College's Kenney Community Center last summer. Watching Gili and the two AAMC veterans she had recruited to help, Cecilia Holodak and Elizabeth Parizh, orchestrate this event was the single most epic math experience of my entire career as a math outreach volunteer! (And I have had many awesome ones, so that is saying a lot!) The photo below shows Gili and Elizabeth with some of their happy treasure hunters and you can learn much more about that treasure hunt in the writeup and photos on Gili's blog here.

Gili leading a Math Treasure Hunt she designed and organized for younger girls at the Kenney Community Center at Union College late last summer . |

After hearing about Gili's very successful local treasure hunt, Ken Fan at Girls' Angle invited Gili to help him lead a much larger treasure hunt at Microsoft New England Research & Development Center as part of a social event ("Games Night") at the Math Prize for Girls at MIT last fall. It was a *huge* hit engaging scores of girls from all over the United States and Canada.

Gili's account of that night is here. Ken describes one fun part of their treasure hunt, Mental Madness, here. In another event, called "Robo-Ape", Gili and Ken asked the girls to compose algorithms to instruct a robotic ape about how to eat a banana. Gili then read their algorithms aloud while Ken played the role of the robotic ape, executing their algorithmic instructions quite literally to great amusement. (You can see a video clip of RoboApe here.)

Ken Fan from Girls' Angle and Gili in the Robo-Ape event |

Attendees at the Math Prize social event included Stephen Wolfram and his 15-year-old daughter Catherine, who was intrigued by the treasure hunt idea that Ken and Gili were leading. Afterwards, Gili and Catherine stayed in touch and worked together to create yet another local treasure hunt back in Schenectady at Union College's Kenney Center in early November, this one with a Halloween theme. You can see a little bit of their treasure hunt in this video (starting at 3:27). Gili described some of their activities in her blog here.

Gili is an outstanding role model, a trail blazer who is creating wonderful road maps that other students can follow as well to create their own mathematical community building events! She is only a tenth grader, but her work thus far exceeds my wildest dreams of what I would have thought possible. And she started out in a small satellite middle school math circle led by Zagreb Mukerjee at a table in the Clifton Park library back in when she was a fifth grader.

Where it all began years ago: a younger Gili (center, back to camera) participating in a small satellite middle school math circle led by Zagreb Mukerjee (standing) at the Clifton Park Library. |

Zagreb is now off in college, but Gili is indeed doing her utmost to "pay it forward" and share the magic of creating vibrant local mathematical communities with younger people in new and innovative ways. And who knows what wondrous activities Gili will--in her turn--inspire the young students with whom SHE is working to do a few years down the road, when it is THEIR turn to pay it forward!

## Thursday, January 24, 2013

### Sign up for AMC10B or AMC12B at Siena College!

Thanks to Siena College Math Department and especially to Professor Mohammad Javaheri (a silver medalist at the 1995 International Math Olympiad) for sponsoring this exciting math contest for high school students in the Capital District. The AMC10/12 is the first in a series of challenging "extreme math problem solving" events that ultimately leads to selection to the US team for the International Math Olympiad! It also opens many other doors as well, including the American Invitational Math Exam, the USA Math Olympiad and USA Junior Math Olympiad. If you are a girl, it is also the entry point to the Math Prize for Girls at MIT next September!

We especially encourage participation from students who enjoy math challenges but who have never even heard of this contest before. High school students interested in joining Albany Area Math Circle sessions to help prepare you to have an enjoyable extreme math experience should contact AAMC advisor Mary O'Keeffe at mathcircle@gmail.com for more information about how to prepare. Please use the form below to sign up to take the contest at Siena on February 20.

We especially encourage participation from students who enjoy math challenges but who have never even heard of this contest before. High school students interested in joining Albany Area Math Circle sessions to help prepare you to have an enjoyable extreme math experience should contact AAMC advisor Mary O'Keeffe at mathcircle@gmail.com for more information about how to prepare. Please use the form below to sign up to take the contest at Siena on February 20.

## Sunday, October 21, 2012

### Congratulations, Zubin!

Zubin Mukerjee, at right, with other members of the Upstate New York All-Star Math Team at the national high school math tournament, ARML, in June |

Zubin Mukerjee, a veteran leader of Albany Area Math Circle who has also organized and led a satellite math circle of his own for younger students, has blazed yet another a new trail for others to follow.

Zubin, a Guilderland High School senior, who is taking advanced classes in math and economics at SUNY Albany, and his co-author, Uthsav Chitra from Delaware, have won semifinalist honors in a prestigious national research contest for high school students. Zubin and Uthsav worked on an original research project in number theory, "Random Involutions and the Number of Prime Factors of an Integer,"under the guidance of a mentor at PROMYS in Boston last summer.

Research presents new challenges as well as new rewards compared to the contest problems with which many math circle students are familiar. When you work on a contest problem, it may be very hard, but you KNOW that you are working on a problem that somebody else has already solved and that there must be a clever elegant solution to the problem. It is indeed exciting to have the Aha! moment when you find the solution to a contest problem, but such moments pale compared to those you can experience in math research, the thrill of discovering an answer to a problem nobody else has ever found before. Sometimes the results are negatives ones, not exactly the ones you were hoping for originally, but even those disappointing results can provide important clues to promising new lines for exploration.

It takes passion, perseverance, and luck to find original new research results, far moreso than in contests. When doing original research, there are no guarantees at the outset that the problem will even have a solution at all, let alone that it will yield interesting results worth sharing with others. Even once the problem is solved, it takes excellent writing skills to write up your research results in a way that will allow others to appreciate the importance and validity of what you have discovered. Zubin's years of helping to write power round solutions for our math circle teams as well as his prize-winning entries in history day competitions have certainly polished his expository writing skills.

Here are the abstract and executive summary for Zubin and Uthsav's research project.

**Abstract:**

*For hundreds of years, mathematicians have tried to find good approximations for the function d(n), which counts the number of prime factors of an integer n. In this paper, we examine*

*using random involutions to approximate d(n) by comparing the number of fixed points of a*

*random involution on F*

_{2}^{2g(n)}to the number of fixed points of a specific involution, τ(n). We find and prove that the expected number of fixed points of a random involution converges, so that d(n) cannot be approximated using this method; moreover, we use this to show that the involution τ(n) is not random, as it has more fixed points than a random involution.**Executive summary:**

*The natural numbers are perhaps the most familiar to humans. They are the counting numbers: 1,*

*2, 3, etc. A divisor of a number is something that divides evenly into that number. For example, 3*

*and 14 are divisors of 42, but 42 is not a divisor of 3 or 14. A prime number is a natural number*

*whose only positive divisors are 1 and itself. The first few primes are 2, 3, 5, 7 ... there are also*

*infinitely many of these. There is a well-known function that returns the number of prime divisors*

*of a number n, given that number. We denote this function d(n). Our goal in this project is to*

*further research on modeling d(n).*

*Our mentor proposed a possible method of modeling d(n) by looking at special functions called*

*involutions that act on the surface of modular curves. In particular, we studied the involution*

*τ*

*(n), which is related to d(n), by comparing it to random involutions. We were able to conclude,*

*through a series of proofs and derivations as well as some graphical analysis using Mathematica,*

*that d(n) cannot be modeled by*τ

*(n) and that, as a result,*τ

*(n) is not random. In other words,*

*there is something special about*τ

*(n) that makes it so we can’t model d(n).*

*The consequences of this result are not yet fully clear. Nevertheless, this result can lead the way to*

*studying other types of involutions, some of which may be able to model d(n). An accurate model*

*for d(n) would be incredible, as it would make finding the prime factorization of large numbers*

*much easier; this would have many applications in cryptography and computer science. Much*

*research remains to be done on involutions though; perhaps one day, a closed-form expression for*

*d(n) will be found through random involutions.*

Zubin and Uthsav's research mentor was Dr. Kirsten Wickelgren, an American Institute of Mathematics fellow at Harvard University. Here is a link to a copy of the background document including the problems she suggested they investigate as well as definitions of some key concepts and a helpful list of the supplementary references with which she initially launched them on their way. If you are interested in understanding more about their work, you may want to take a look at those references yourself. Students who have not yet studied much number theory will also find the

*Art of Problem Solving'*s textbook on introductory number theory very helpful. [Added later: Zubin also passed along another recommendation of a classic number theory book, Hardy & Wright's

*Introduction to the Theory of Numbers,*endorsed by PROMYS Director Glenn Stevens as "clear and concise." Zubin also notes that Hardy & Wright cover many topics in number theory in their book, some relevant to their project and some not.]

You will also note that Zubin and Uthsav used Wolfram

*Mathematica*computer software to help create graphs to give them insights into their problem analysis. Thanks to Wolfram's sponsorship of contests such as American Regions Math League (ARML), Harvard-MIT Math Tournament (HMMT), and Princeton University Math Contest (PUMaC), Zubin and all our veteran math circle students who have participated in one or more of those contests have received free student licenses to use this very powerful software. Those licenses will remain valid as long as they are students, including college and graduate school years ahead.

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