Frequently Asked Questions About Albany Area Math Circle

(Updated September 2025)

What is Albany Area Math Circle?

Albany Area Math Circle (AAMC) is a group of high school students from all over the Capital District who meet for two hours roughly biweekly on Sunday afternoons from 4 to 6 p.m. to collaborate on exciting and challenging math problems together.  Our meeting schedule generally runs from early November through early May.

You might call it “Extreme Math.” The problems are “outside-the-box.” Students need to come with enthusiasm for working hard and cheerfully making mistakes (which are inevitable—everyone makes them, even the veterans who have been part of the circle for many years. The adults advising the group cheerfully admit that they make mistakes as well—it is part of the fun of the experience.) The key is for all members of the circle to share their half-baked approaches and ideas and discover that the whole really is more than the sum of the parts. The essence of the fun is a “Shared Aha! Experience” when several students finally crack a problem together.

Who is eligible to join AAMC?

High school students from all around the Capital District are welcome to become part of our community. There is no qualifying test for high school students. You will need to register by joining our email list before your first meeting. (See instructions below for how to do this.)

The only requirements for a high school student member are enthusiasm for working hard and cheerfully on challenging math problems, a willingness to make mistakes, to celebrate your mistakes and learn from them, to share both cool insights and half-baked solution approaches with other students.

We expect all members of AAMC to work with one another in a generous, kind, and helpful spirit. We expect all students to behave in a way that will contribute to an enthusiastic and productive learning community.

Are younger students eligible to join AAMC before ninth grade?

The problems we work on at AAMC at our regular Sunday meetings are very challenging, even for advanced high school students already taking college courses. Furthermore, our meetings are two hours long, which is a long time to work on hard math problems!

However, AAMC's high school students provide many outreach efforts for younger students who enjoy math challenges.

We encourage our high school student members to serve as assistant coaches in MATHCOUNTS programs at nearby middle schools. Over the years, AAMC members have enjoyed serving as student coaches for MATHCOUNTS programs at Acadia, Doyle, Farnsworth, Gowana, HEEG, Iroquois, Koda, and Van Antwerp Middle Schools. If your middle school would like a student coach to assist the teacher or other adult sponsor coach, AAMC would be happy to assist in recruiting one.

Some of our high school student members serve as coaches of satellite middle school math circles.

We encourage local schools to offer the AMC contests, starting with the AMC8 for students in eighth grade and below, which is offered in November. Middle school students who achieve Honor Roll scores on the AMC8 are encouraged to take the AMC10 or AMC12 contests in November. These high school contests are extremely challenging, but they can be good learning experiences for a few younger students who are ready to take the next step beyond middle school math contests. Students who score high on the AMC10 or AMC12 contests may qualify for the American Invitational Math Exam (AIME) given in March.

With the exception of our special outreach activities described above, our regular AAMC meetings are organized for high school student members. Very rarely, middle school students who have qualified for the AIME may also be invited to join our regular meetings. This is very much the exception, rather than the rule, and younger students should not be discouraged if they find the AMC preliminary exams extremely challenging. Some of our strongest high school members only qualified for AIME after several years of trying.

There are many excellent math enrichment resources available for younger students at libraries, bookstores, and on the Internet.

Do AAMC members participate in any math contests? Yes, AAMC students have joined together to form teams that have competed in a number of math contests over the past five years. AAMC teams have done very well, but you should know that participating in math contests is strictly optional for AAMC members. We realize that official competitions are not everyone's cup of tea. There is no requirement to participate in official math contests and students who just want to join in the fun of working hard on challenging problems at our meetings are very welcome! One important note: if your school has its own team for a particular competition, you should compete on your school's team for that particular contest, NOT on AAMC's team. We want to support those schools that choose to field their own teams.

How do I sign up for the AlbanyAreaMathCircle email list?

Step 1) High school students and parents should go to https://groups.google.com/g/albanyareamathcircle/ and request to join the group. If you have difficulties in signing up, please email AAMC Advisor Alexandra Schmidt at aleph@stanfordalumni.org for troubleshooting assistance. Please send a separate message originating from each email address where you would like to receive math circle emails. So, for example, if Sarah and both of her parents all want to subscribe to our email list, they should each send an email to the above address from their respective emails. The body text of your email can just briefly state your name, school, and grade (or your student's name school and grade) so we can recognize you as a math circle member or parent and approve your subscription. Once you are approved for subscription, you will automatically get information about problems to print out and bring to meetings, how to sign up for math contests with us, and many other important details! Step 2) Please use the form at this link to give us all the essential contact information for your student.

Where and when do you meet? This year, we are again meeting on Sundays afternoons from 4 to 6 p.m. in Troy. Members who join our googlegroups email list will get information about the exact location and parking/entrance directions.

Who are the AAMC advisors? The current lead advisor/head coach for Albany Area Math Circle is Ms. Alexandra Schmidt.

The founding AAMC Advisors in 2001 were Mary O'Keeffe and Mukkai Krishnamoorthy. They have served as volunteer advisors to the group since its founding in 2001, and they still help out.

Is there a cost to joining AAMC? We are an all-volunteer organization and we do all we can to keep costs minimal. We also try to keep costs down at travel contests. Entry fees are generally moderate ($10 per student is typical.) advisors coordinate information so parents can make arrangements with one another to carpool to distant contests. Some contests can be done as day trips to hold down overnight lodging costs. Other contests are held on college campuses and, with advance notice, it is sometimes possible for students to stay with college student hosts in the dorms at minimal cost. In most years, the statewide NYSML tournament arranges host families who are willing to accommodate students traveling from a distance. (Last year's lodging arrangements for NYSML were very unusual, due to the NYC location.)

Where do AAMC members live? Albany Area Math Circle brings together students from a wide variety of communities. Our membership has changed year to year. Students and alumni have hailed from all over the Capital District including the following communities: Albany Averill Park Ballston Lake Bethlehem Clifton Park Coxsackie Cropseyville Delmar Fort Edward Guilderland Latham Niskayuna Saratoga Springs Scotia-Glenville Schenectady Troy Watervliet AAMC members have come from the following schools: Albany High School, Averill Park High School, Bethlehem High School, Coxsackie High School, Guilderland High School, Niskayuna High School, Schenectady High School, Scotia-Glenville High School, Shaker High School, Shenendahoah High School, and Troy High School. We also have homeschool students and students from the Albany Academies, Doane Stuart, and Emma Willard. We are happy to build bridges of collaboration among students who might not otherwise have an opportunity to work together.

MATHCOUNTS Congratulations!

I am delighted to share the exciting news that two Albany Area Math Circle members placed among the top four individuals at last week's New York State MATHCOUNTS championship contest, earning the right to represent our state at the National MATHCOUNTS Competition in Orlando Florida in May.  David Li from Utica was the New York State champion and Rowechen Zhong from Iroquois was the fourth highest scoring individual, and his school team (pictured below) was second place in the state.

Iroquois Middle School team
Steve Schmidt (coach), Stephanie Vernooy (coach),
Harini Prabaharan, Jeffrey Huang, Christine Lee, [name redacted], Rowechen Zhong

Congratulations to the entire Iroquois Middle School's team and coaches for their outstanding second place finish at the state contest last weekend.  Veteran head coach Professor Steve Schmidt is my colleague in the economics department at Union College as well as the husband of AAMC advisor Ms. Alexandra Schmidt. Steve headed a strong coaching team that included Professor Stephanie Vernooy (from Siena College's Biology Department) and AAMC student coaches Gabriel Kammer and Ian Vernooy.  AAMC alumni Gideon Schmidt and Patrick Chi served as student coaches in prior years helping veterans on the Iroquois team build a firm foundation in their earlier years of middle school, as did prior adult coach Ambady Suresh.

Professor Schmidt will coach the New York State team at the National MATHCOUNTS Competition in May and will accompany David, Rowechen, and their two teammates to Orlando.

Congratulations again and best wishes to all!

Intriguing guest speaker: Professor Ivana Alexandrova

This spring, our math circle was fortunate to have a special treat:   Professor Ivana Alexandrova from UAlbany gave our high school students a guest presentation full of fascinating problems in trigonometry.  Professor Alexandrova grew up in Vratsa, Bulgaria, which has a rich and venerable tradition of math circles, and she herself participated in math circles as a student before coming to this country for college and graduate studies.  As a graduate student at UC-Berkeley, she met another mathematician from Bulgaria, Professor Zvezdelina Stankova, who leads the Berkeley Math Circle, and she was happy to learn that math circles have started to establish themselves in this country as well.

Professor Ivana Alexandrova with math circle student leaders Gili Rusak and Phil Sun after her intriguing talk.  Gili and Phil are graduating seniors and longtime members of Albany Area Math Circle.  Like Professor Alexandrova, they have grown up as members of a math circle that helped to develop their passion for problem solving and they have been sharing their passions and talents with younger students.  Next year Gili will be studying at Stanford and Phil will be studying at MIT.

Professor Alexandrova, whose research focuses on partial differential equations and mathematical physics, chose to discuss trigonometry in her presentation, because it is a beautiful, highly useful, and often misunderstood subject which is unfortunately somewhat neglected in conventional American mathematics education.  Many college professors share her wish that their students had stronger backgrounds in trig.  Under her guidance scary looking trig problems that might initially make your eyes glaze over sparkled and came to life as she developed beautiful ways to approach them.  Her trig problems were indeed very intriguing!

You can see some of her presentation in the whiteboard shots below.  She has also created a very nice website for high school students with weekly problems designed to draw less experienced students into the world of intriguing math, with a few algebra problems mixed in among the trig problems.  You can find her problems and solutions here.   The current problem is very nice:  Solve 1 + sin x + cos x < 0.

Math Circle plans for the 2013-2014 school year: new time/day/location

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.

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

Does math class make your child feel like a hamster in a cage stuck in a wheel of an endlessly repetitive "spiral curriculum" with little to challenge or inspire her?   If you answered yes, then this book could provide a much-needed breath of fresh air.

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.

Sumer is icumen in,

Lhude sing cuccu!

Groweth sed and bloweth med

And springth the wude nu,

Sing cuccu!

Enjoy your summer!  Parents may find they too enjoy learning some new mathematical insights if they talk about these problems with their children.  It is great for students to discover that sometimes they can figure out answers to problems that stump grownups!  As I have discovered myself, time and time again, when working with my own children as well as other people's children in my math outreach activities, while it may be humbling for me, it is empowering and exciting for children when a flash of insight enables them to climb a mathematical mountain before I do. 

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

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 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.

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, 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.

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.

Math Prize for Girls @MIT participants enjoy the extremely fun yet challenging math treasure hunt Kan Fan and Gili ran at a "Games Night" social event at Microsoft New England Research and Development (NERD) Center.

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!

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.

 

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 F₂^2g(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.

Math Circle travel contest dates this year

Math contests are a fun (but entirely OPTIONAL) part of our Albany Area Math Circle experience.

To help our members plan ahead, I will describe the entire calendar of contests in which members can participate below.  I have broken them down into two categories: travel contests and locally administered contests.  This post describes the travel contests we plan for this coming year.  My next post will describe the locally administered contests we plan.


Harvard-MIT November Tournament (HMNT):  
Saturday November 10 at Harvard University in Cambridge, MA   
This is an ideal travel contest for our younger veterans who have done at least NYSML in the past, but who have not yet done much proof-writing.   Note that participating in this contest rules out participating in PUMaC the following weekend.  It also rules out participating in the (much harder) February 2013 HMMT.  If you are seriously considering this contest but not yet sure of your schedule, please email Mr. Babbitt ASAP to let him know of your tentative (or definite) interest.
HMNT November Cost per student:  $10 registration fee plus modest costs for lunch.  AAMC can put parents in touch with one another to arrange carpools so that parents can share gas and tolls.  Math contests at Harvard and MIT have been done as a daytrip by AAMC mathletes who were comfortable getting up early and sleeping in the car.   If your parents agree, it may also be possible for students to arrange to stay overnight in a Harvard or MIT dorm with a college student host or hostess.   (You would need to bring a sleeping bag, and you will probably be sleeping on the floor.  Note that dorms can be noisy on Friday nights!)  Parent drivers may want to work out arrangements to share hotel rooms (unless they are driving as a day trip.)   Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.


Princeton University Math Contest (PUMaC):  
Saturday November 17 at Princeton University in Princeton, NJ 
This is a very challenging contest which includes a collaborative team round.   Only experienced veteran students should consider this contest.  It is considerably harder than the AIME, so students who have not yet taken the AIME should not consider it.  But even the AIME is not sufficient preparation, because the team round requires mathematical expository skills.  The team round is proof-based and essay style.  Students need to be prepared to work collaboratively (and remotely) on a google document from Nov 10 through November 16.  In addition, there will be mandatory on-line practice in October for all team members on our AoPS forum.  Non-team members are also welcome to join the on-line practice forum.  Our Math Circle veteran leaders, Matt Babbitt and Zubin Mukerjee, will provide further information about what is expected of team members on the AoPS forum.  If you are seriously considering this contest but not yet sure of your schedule, please email Mr. Babbitt ASAP to let him know of your tentative (or definite) interest.
Cost per student:  $12.50 registration fee (which includes lunch).  AAMC can put parents in touch with one another to arrange carpools so that parents can share gas and tolls.  Parent drivers will probably want to work out arrangements to share hotel rooms.  Due to the distance involved as well as the need to finalize the power round with a final team meeting in Princeton on Friday evening, this event is NOT doable as a day trip.  Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

Harvard-MIT Math Tournament (HMMT)
Saturday February 16, 2013 at MIT in Cambridge, MA
This contest has always been extremely hard, and we expect that trend to continue to unprecedented epic levels of difficulty this year.  Only those with extreme tolerance for mathematical endurance should consider this event.  However, mathematical strength is not enough to be selected for this team, since collaboration with other team members will be very important.  It is very important that all members of our HMMT team know one another's strengths and weaknesses so that the "whole can be more than the sum of the parts."  It is also important that all team members encourage one another when the going gets tough, because--believe me!--it will get tough!

Our advanced veteran students who wholeheartedly participate in fall math circle activities including our PUMaC on-line practices on the AoPS forum and regularly attend (most of) our Friday meetings with an extremely positive and supportive spirit that encourages others not to give up when the going seems hopeless will be given first preference in team selection.   All other factors being equal, seniors will be given preference over younger students.   For younger students, participation in previous years' HMMT on-line events or travel events will be a plus factor.  All other factors being equal, younger students who have done well in prior year on-line HMMT events will be given preference over those who attended the travel event and did equally well.

Cost per student for Feb HMMT:  $10 registration fee plus modest costs for lunch.  AAMC can put parents in touch with one another to arrange carpools so that parents can share gas and tolls.  The event can be done as a daytrip if your mathletes are comfortable getting up early and sleeping in the car.   If your parents agree, it may also be possible for students to arrange to stay overnight in a Harvard or MIT dorm with a college student host or hostess.   (You would need to bring a sleeping bag, and you will probably be sleeping on the floor.  Note that dorms can be noisy on Friday nights!)  Parent drivers may want to work out arrangements to share hotel rooms (unless they are driving as a day trip.)   Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

New York State Math League (NYSML)
Saturday April 13, 2013 at Biram Hills  High School in Westchester County
NYSML is an ideal "first" travel competition for our students.  It is the statewide math championship for high school students.  It is held in a different place each year.  We (Albany Area Math Circle) were the host league in 2010, Suffolk County in eastern Long Island hosted in 2011, Southern Tier was the host league in 2012.  The 2013 host will be the Westchester-Putnam Math League.    All high school students who attend meetings regularly and demonstrate mathematical and behavioral maturity as well as enthusiasm and persistence are welcome to join our NYSML teams.  Unlike the other travel tournaments, we expect to be able to take as many 15-person teams as we would like to NYSML.  (In 2010, we had three teams with almost 45 students.  If a team is a bit short of 15 students, we can use alternates from other teams and vice versa at NYSML.)

Cost per student for NYSML:  Unknown at this time, but probably around $20 per student (including lunch).  It is a potential day trip (for mathletes who don't mind getting up early and sleeping in the car.)   Some parents and mathletes may prefer to stay in a hotel.  As with all travel contests, we will facilitate parents getting in touch with each other to make their own carpool/travel arrangements.   Although we will put parents in touch with each other to facilitate whatever arrangements seem most suitable to them, it is entirely up to the parents involved to work out mutually satisfactory travel arrangements.

American Regions Math League (ARML)
Saturday June 1, 2013 at Penn State (depart Albany morning of 5/30 and practice with team on 5/31 at Penn State, so students need to miss several days of school for this.)
ARML is the national high school championship event.  It has a format similar to NYSML but the problems are harder.  Albany Area Math Circle students can apply to be on the Upstate NY ARML team.   You will want to wait until you have your AMC, and (if applicable) AIME and NYSML scores before you apply to the team.  NYSML experience and performance will also be a big plus for ARML team selection.

Important info about conflict with SAT testing date:  if you are thinking of doing ARML, please plan ahead for the SAT conflict.  June 1 is an SAT administration date.  It is expensive and time-consuming and may not be possible to arrange a makeup date.  It is MUCH better if you plan to take your SAT subject tests in May rather than June, and/or your SAT reasoning tests in January or March rather than June.  The SAT schedule for this coming year is here.

Cost per student for ARML:  Unknown at this time, but in recent years the cost has been around $300 to $325.  That included a chartered bus from Binghamton to Penn State and back as well as two night's housing in Penn State dorms and some, but not all, meals.  Students also needed about $25 to buy additional meals on their own.  Thanks to Cecilia and Gili's moms efforts in fundraising the NYSSPE donated funds that helped to offset about $100 of the costs per student.

In the next post, I will describe the contests that we expect to be offered locally, without the need to travel.

Upstate NY All Stars!

Albany Area Math Circle members of the New York State All Star Math Team thank the New York State Society of Professional Engineers for their support:  
Yujun (Etheal) Chen, Emma Willard School
Sherry He, Emma Willard School
Gili Rusak, Shaker High School
Jien Ogawa, South Colonie High School
Cecilia Holodak, Niskayuna High School
Matt Babbitt, heeg
Zubin Mukerjee, Guilderland High School
Ziqing (Bill) Dong. Farnsworth Middle School

Complete 2012 Upstate NY AllStar Math team roster for 2012 ARML, including students from all over Upstate New York (defined as New York State minus {New York City, Nassau County, Suffolk Count})

Matthew	Babbitt	heeg
Jemmin	Chang	Somers
Yujun	Chen	Emma Willard
Yiqun	Cui	Brighton
Matt	DeCross	Pittsford-Sutherland
Pranav	Devarakonda	Brighton
Matt	Dobbins	Corning West
Ziqing	Dong	Farnsworth
Ryan	Gao	Brighton
Huajun	Gu	Corning West
Jeff	Guo	Penfield
Sherry	He	Emma Willard
Cecilia	Holodak	Niskayuna
Doug	Knowles	Churchville-Chili
Jeremy	Koob	Corning West
Benjamin	Lei	Arlington
Devin	Li	Corning West
Allen	Liu	Penfield
Ben	Lowenstein	Pittsford-Sutherland
Dongze	Lu	Harley-Allendale-Columbia
Evan	Lustick	Canandaigua
Zubin	Mukerjee	Guilderland
Jien	Ogawa	South Colonie
Jordan	Roeder	Canandaigua
Gili	Rusak	Shaker
Perry	Wang	Corning West
Felix	Weilacher	Penfield
Yunmei	Zhang	Harley-Allendale-Columbia
Simon	Zheng	Somers

Guerrilla Math venues?

Schenectady--and more broadly, the Capital District, have many wonderful free outdoor activities.  They are fun family places to go and enjoy our beautiful summers.  And they are great places for you and your families  to do Guerrilla Math!  Just bring some sidewalk chalk or a portable whiteboard and markers, and your ideas for fun math explorations to share with families waiting for the event to begin or during intermissions, afterwards. 

Here are few links I got from a colleague at Union College.  If you can suggest other links, please post in the comments below.

Central Park (Music Haven) Schenectady summer concerts

Jazz on Jay Street

Freedom park summer concerts (Scotia)

Other Schenectady events

Guerrilla Math Circle at the Youth Peace Rally

It is a beautiful day for more sidewalk chalk Guerrilla Math Circles.  I will be doing it at today's Youth Peace Rally at Jerry Burrell Park in Schenectady.

As I wrote in my last post:

Jerry Burrell Park is a place that is near and dear to my heart, because one of my former students at Union College, Jeremy Taglieri, spent a year working along with two friends to raise $84,000 in funds and recruiting 140 volunteers to renovate the park.  They called their project Project SKIP, where SKIP = "Schenectady Kids Imagine and Play."  Jerry Burrell Park is also the site of a SICM free lunch site, and in fact, volunteers from my own church will take a turn serving lunch for a week there later this summer. 

All math circle families, siblings, friends, and neighbors are cordially welcome to join us.  But--as I also wrote in my last post--I am mindful that not all parents will be comfortable with their families coming to this neighborhood.  It is a decision that should be made with full information--this is one of the highest crime neighborhoods in our area, and in 2010, just a few weeks after the playground's renovation, a 17-year-old shot and seriously wounded an 11-year-old boy and his 18-year-old older brother.  Each family needs to make a choice they are comfortable with.

But today, I choose to join those who hope to bring and keep peace in this community.  And I will bring what I have to share--the beauty and excitement of creating and discovering mathematical patterns.

Directions:  Jerry Burrell Park is located at the corner of Hamilton and Schenectady Streets in the Hamilton Hill neighborhood of Schenectady.

Pascal's triangle in sidewalk chalk

First 15 rows of Pascal's triangle (mod 2) in sidewalk chalk

Imagine that you throw a dart at Pascal's triangle.  What are the odds you will hit an even number?

Paul Zeitz raised this fascinating question at the MSRI Great Circle's workshop three years ago.

It's a fun one to explore in sidewalk chalk--perfect for Guerrilla Math Circles.

A good way to start thinking about this problem is simply to begin coloring in the triangle--and what could be more fun that the idea of creating a really large one with sidewalk chalk?  And engaging any curious passersby in helping you.  How big can we make it if we get lots of folks helping us?  (And of course, we could also start some people on exploring Pascal's triangles with multiples of other prime numbers colored in.)

I started one the other day in Schenectady's Central Park and I will be back there this evening (Friday June 8) to do more.  There is a nice large stretch of blacktop that is blocked off from automobile access.  It is adjacent to Tiny Totland and not far from the Music Haven (site of many wonderful free concerts) and also near the picnic pavillion area where Schenectady Inner City Mission serves free lunches in the summer.  Central Park is the "crown jewel" of Schenectady's park system, designed by Frederick Olmsted and with a world class rose garden.  To get to where I will be tonight, enter Central Park via the main entrance off Central Parkway and then turn to the left when you get to the lake.  Park in the parking lot near Tiny Totland and look for me in the blacktop area between Tiny Totland and the lake.

On Sunday afternoon, from 1 to 5 p.m. I will be doing more Guerrilla Math Circles at a different park in Schenectady, Jerry Burrell Park.  Jerry Burrell Park is a place that is near and dear to my heart, because one of my former students at Union College, Jeremy Taglieri, spent a year working along with two friends to raise $84,000 in funds and recruiting 140 volunteers to renovate the park.  They called their project Project SKIP, where SKIP = "Schenectady Kids Imagine and Play."  Jerry Burrell Park is also the site of a SICM free lunch site, and in fact, volunteers from my own church will take a turn serving lunch for a week there later this summer.

I invite Albany Area Math Circle families (students, parents, siblings) to join me in sharing the fun, beauty, and awesomeness of mathematics with a wider world, a world of folks who have yet to discover how much delight can be had in exploring the wilds of mathematics.

All of you can do these activities wherever your parents feel comfortable with your doing them.  I grew up in a city myself, and have always been comfortable having my children play in Schenectady parks when they were growing up, with--of course--sensible precautions taken about when and where to go.  Personally, I am comfortable being in city parks when SICM volunteers from churches all over the area are serving free lunches, but not when they are mostly deserted.  And this Sunday's event at Jerry Burrell Park is part of a larger event organized by city officials.

Nothing is ever 100% safe--and there are risks to just sitting in your own living room (a car might come crashing through the wall), but I recognize that these are decisions that math circle parents needs to make for their own children.  I feel compelled to point out that--sadly--just a few weeks after the renovation of the Jerry Burrell Park in 2010, a 17-year-old boy shot and seriously wounded an 11-year-old and an 18-year-old boy who were walking to the park from just a block away.

But--remember that you and your sons and daughters can also do these same activities WHEREVER you happen to be--on the sidewalk in front of your own home or the park just down the street from you in your own neighborhood.

No matter where you are comfortable with participating in our outreach activities, you can work with us to brainstorm ideas and post links to photos in the comments below.  Albany Area Math Circle members can also post about their experiences in our private discussion forum on the Art of Problem Solving.

Home of the math champions: Upstate New York!

Upstate NY All-Star Math Team celebrates the announcement that Allen Liu is the national high school math champion!

Hundreds of mathletes and coaches applaud Allen as he walks down to center court at Penn State's Bryce Jordan Center.  Allen's victory is all the more impressive because he is only an eighth grader.  He attends Bay Trails Middle School in Penfield, New York, but he has already been excelling in math classes at the University of Rochester.  He is currently attending the Math Olympiad Summer Program, along with high school students on this year's International Math Olympiad team and others considered strong prospects for future year teams.  As a USA Math Olympiad  qualifier every year since the fifth grade and a top-10 finisher at ARML since sixth grade, Allen has already compiled an extraordinary track record, yet he is an outstanding example of modesty and a generous spirit always willing to share what he knows with others.

Upstate NY All-Star Coaches Jen Vibber, Brenda Munch, and Tom Zink beam with pride as Allen becomes the newest member of their Upstate NY dynasty of national champions. No other ARML team anywhere in the country (or world!) can boast about so many national champions.  ARML went national in 1984. Since that time, 26 distinct individuals have won ARML.   Of those 26 champions, four are from the Upstate NY team:

Robert Kleimberg (1992)
Jeremy Bem (1993)
Aaron Pixton (2004)
Allen Liu (2012)

Coach Jen Vibber has been Allen's local coach throughout his middle school years, as well as a member of the Upstate NY Math Team coaching staff.  She is also the incoming head coach for the Upstate NY Math Team.  That's outgoing Upstate NY head coach George Reuter looking over her shoulder in the photo!  With the foundation built by George, Jen, and the other dedicated veteran coaches on the Upstate NY staff, we all know the Upstate NY team is in great hands going forward.  Thanks to all the tremendously dedicated coaches for the many hours of tireless work they put into creating an awesome community experience for mathematically passionate kindred spirit students all over Upstate NY.   It is obviously a labor of love for each and every one of them, as they are all volunteers.   They are outstanding teachers who inspire their mathletes with their character as well as their knowledge of problem-solving methods.

2012 Upstate NY Math Team Coaching Staff:

George Reuter (MCML/Canandaigua Academy) 
Doug Becker (MCML/Gates-Chili HS), Treasurer
Brenda Munch (MCML/Brighton High School)
Patty Pragel (MCML/Greece Arcadia High School)
Jen Vibber (MCML/Penfield High School)
Glen Stephenson (Corning West High School)
Tom Zink (AAMC/MicroStrategy)
Bill Babbitt (AAMC/RPI)
Rita Biswas (AAMC/UAlbany)

(If you would like to learn more about the "Parent Function" cited on the Upstate NY team t-shirts, check out Pat Ballew's blog post describing the fascinating history of the term's usage.)

NY team scales new heights at MATHCOUNTS Nationals

The New York State MATHCOUNTS team raised the bar to the highest heights in recent memory with a fourth place finish at the National MATHCOUNTS competition held in Orlando, Florida last week.  The photo above shows the team celebrating on Expedition Everest at Disney World after the contest.

Three-year team veteran Allen Liu from the Rochester area reached the National Countdown for the third consecutive year, the first New Yorker ever to accomplish this feat, and reached the semifinals for a top four individual finish.

Albany Area Math Circle's own Alex Wei had one of the best finishes ever for an Albany Area mathlete, coming in a very impressive 18th place among the 224 national finalists.  Alex's first MATHCOUNTS coach, who launched him on his way back when he was in sixth grade, was another former national mathlete from the Albany Area, Dave Bieber.  (Dave, in turn, had been coached by yet another former national mathlete from the Albany Area, founding AAMC member Alison Miller, when he was in sixth grade!)

For the first time anyone can recall, all four members of the team, which also included two-time veteran Peter Mizes from the Rochester area and Calvin Lee, the Manhattan MATHCOUNTS chapter champion, all earned top 56 recognition.  Most impressively of all, they were able to come in very strong on the collaborative team round, allowing their team to rank higher than the traditional powerhouse states of California and Texas.

Because of the very large distances separating the mathletes and the fact that there is no travel budget to enable the team to practice together as a team prior to nationals, the students seized the initiative, took ownership of their learning, and organized their own on-line collaborative practices prior to nationals, without requiring adult assistance.  Essentially, they coached themselves.  The team clearly made New York State proud in many ways--their friendly collaborative work together carried on a fine tradition to new heights and surely will inspire future mathletes from our state to new heights, as they stand on the shoulders of these mathematically collaborative and resourceful giants.

If you would like to see the problems they tackled, you can find them posted on the Art of Problem Solving discussion forum here.   One of my favorite problems from this year's national contest is here--it is a beautiful illustration of how probability problems can often be solved by viewing them as geometry problems.