excellent new issue of Girls Angle bulletin

The bulletin is put together by the Girls Angle program in Cambridge, MA, where the mentors include Harvard and MIT students and faculty working with girls in grades 5 through 11. The bulletin is chock full of delights for the younger members of Albany Area Math Circle, for our older students looking for ideas to mentor them, and for all those who are young at heart mathematically, no matter what their gender.

Check their latest issue out here!

Harvard freshman Amy Tai has written a beautiful and very engaging exposition of the properties of the medians of a triangle, which begins on page 2.

Hah! I can see some eyelids drooping already.

The medians of a triangle....how exciting a topic could that be? Like watching paint dry, some of you are probably thinking to yourselves.

Too many American high school students think of geometry as a dull subject to be gulped down like cod-liver oil before moving on to more advanced topics. Amy's article makes the topic come alive as she invites the reader into the exploration along with her.

Amy's article is a superb model for younger students who really want to develop their geometric intuition AND learn how to write crystal-clear mathematical expository prose. Her illustrations make a potentially dull topic spring to life in a memorable way.

Our Albany Area Math Circle high school students who are mentoring younger students can gain some very valuable insights and ideas from reading Amy's article and related pieces in the bulletin.

The Girls' Angle bulletin also contains more articles that generalize on the idea of the median. A median of a triangle is a line passing through a vertex that cuts the area of the triangle in half. Determining the analogs for more complicated polygons turns out to be a very fascinating activity. Try finding those lines for the shapes on page 15. They start out easy, and then require more and more ingenuity as you move down the page. They are sort of like potato chips--bet you can't eat just one!

The latest Girls Angle bulletin also features a story written by MIT senior Maria Monks about the adventures of a young mathematician named Emmy Newton. Maria writes in a way that invites the reader to solve the puzzle along with--or even before--Emmy does. Congratulations to Maria, who has won many honors, including Honorable Mention for the 2010 AMS/MAA/SIAM Morgan Prize for outstanding research in mathematics by an undergraduate. She will be giving a talk on her research at the Joint Math Meetings in San Francisco next month.

Who can resist the mathematical treasure hunts included in the pages of this issue? If you think you can solve the one on page 10, you are invited to send your solution into the journal. You could, of course, take the easy way out and use a calculator to solve it, but students may discover many beautiful ideas if they take their time and think through the problem in a different way. Our middle school math circle students who heard me talk about bimal last fall may especially enjoy thinking through a part of the problem that intially looks long and computationally messy--but it isn't really, once you think about the meaning of bimal(bimal is the binary analog of decimal notation.)

Kudos to the creators of this excellent bulletin. There's much more packed in the issue that looks fascinating and promising as well. Back issues are also available on-line in their archive here. There are many more treasures there.

the generosity of math team captains

Brian Hamrick, cocaptain of the legendary powerhouse Thomas Jefferson High School math team writes:

Winning is great fun when it happens, but it shouldn't go to your head and you should be focusing on learning first, winning second (if even that). I have made the math team wiki public for that reason: I would rather have our competitors know much of what we know and give us a good contest than have a trophy on my shelf.

Thomas Jefferson has made available a huge set of great resources here.

Winter break suggestions

Winter break is a great time for students to discover the joys of recreational math reading. Here are some suggestions which can delight students from middle school through the rest of their lives. The books below are classics. Many should be available in public libraries and/or inexpensively in used copies on the Internet.

Mathematical People and More Mathematical People have many delightful stories of the lives of noted 20th century mathematicians. Anyone who thinks mathematicians lead boring lives is in for a surprise when they read these interviews. Magic card tricks, chess puzzles, juggling, trampolines, and Mad Magazine all figured into the lives and mathematical development of one or more of these colorful characters.

Richard Feynman was a physicist, but his funny autobiographical works (What do you care what other people think? and Surely you are joking, Mr. Feynman.) make many references to his unorthodox mathematical education.

Raymond Smullyan's mathematical logic puzzle books (What is the name of this book? and The Lady or The Tiger and many more treasures along those lines) are also great fun.



Martin Gardner's Aha! Insight and Aha! Gotcha! books are great "entry-level drugs" into his abundant collection of recreational math books. You can see our well-worn copy above--the wear and tear reflects its extensive reading and re-reading over the years.

Alexandra Schmidt, math teacher and MATHCOUNTS coach at Hebrew Academy of the Capital District, recommends The Man Who Counted.

If your library uses the Dewey system, try browsing around 510 or 793. If your library uses the Library of Congress numbering system, try browsing around the QA93 section.

You are likely to find some real treats.

Happy 2010

Here are some currently known contest dates to mark into your calendar for winter and spring of 2010. Clicking on the links below will give you more details for each contest, starting with the physics olympiad on January 23 and running through ARML in early June.



In advance of each contest, we will send out an email asking that interested students discuss the contest with their parents and use our signup form to sign up for that contest.

If any of these contests look interesting to you, be sure to talk to veteran math circle members and advisors for more details.

One particular concern for younger and newer students is the AMC10/12 choice in February. We will talk about this choice in our upcoming meetings.

Does your dog know calculus?

Friday night, Prof. Orzel talked to us about how to teach your dog physics.

Perhaps Elvis, pictured above, would make a good student of physics. His owner, Prof. Tim Pennings of Hope College, has verified experimentally that Elvis "knows" calculus, or at least that he behaves as if he knows calculus.

Prof. Pennings reached his conclusions by taking measurements of Elvis running and swimming to fetch balls he threw into Lake Michigan, but he also suggests that you could conduct the same kind of experiment on your dog by throwing a stick into some deep snow near a cleared sidewalk.

We confess that although he made good choices, Elvis does not know calculus. In fact, he has trouble differentiating even simple polynomials. More seriously, although he does not do the calculations, Elvis’s behavior is an example of the uncanny way in which nature (or Nature) often finds optimal solutions. Consider how soap bubbles minimize surface area, for example. It is fascinating that this optimizing ability seems to extend even to animal behavior. (It could be a consequence of natural selection, which gives a slight but consequential advantage to those animals that exhibit better judgment.)

Finally, for those intrigued by this general study, there are further experiments that are available, other than using your own favorite dog. One might do a similar experiment with a dog running in deep snow versus a cleared sidewalk. Even more interesting, one might test to determine whether the optimal path is found by six-year-old children, junior high aged pupils, or college students. For the sake of their pride, it might be best not to include professors in the study.

Do dogs know calculus? College Mathematics Journal

Last night was EPIC!

Thanks to George Reuter for writing a great contest!

Mr. Reuter, when do you sleep? For those who don't already know, Mr. Reuter is a math teacher at Canandaigua Academy in western New York, and he is a father of five (including toddler twins!), and he is head coach of the Upstate NY Math Team, and he is also Vice President of NYSML. He also manages to maintain an awesome sense of good humor (check out this facebook page dedicated to collecting his funny remarks.)

He is also a personal inspiration to me. As I tried to emcee the contest for 45 students from the Albany area last night, I wondered how in the world Mr. Reuter managed to keep his cool and maintain his sanity while emceeing the NYSML state tournament last April for over ten times as many students from all over New York. He made it look easy. I know he will be an awesome emcee for this year's NYSML state tournament, which we will be hosting at UAlbany on April 10, 2010.

Thanks to NYSML Fall problem reviewers Mike Curry, Beth Schaffer, and Tom Zink, who did an excellent job of meticulously reviewing the problems. As noted before, Beth and Tom are both alumni and former captains of Albany Area Math Circle, who have continued to contribute their talents and energies to younger students in Albany Area Math Circle and Upstate NY ARML students, indeed to students all over the country and the world, in a variety of ways.

Thanks to Albany Area Math Circle advisor Professor Moorthy, who did an outstanding job of organizing our scoring room last night, with an awesome team of parent volunteers: Shuang Lin, Naomi Ogawa, Michal Rusak, and Katherine Scheib! They did an exceptional job of staying on top of a steady flow of answer sheets steadily streaming into their hands. It's really an amazing amount of papers to keep track of in a very short period of time, and their work was exemplary and efficient.

Thanks also to our AWESOME math circle alumni who were home on their college winter breaks and who returned to help out with administering the contest: Drew Besse, Liz Simon, Susanna Todaro, and Lindsay White. Without your help, and without the help of parent volunteers Rita Biswas and Anil Tolpadi, the contest would have been complete and utter chaos.

Thanks to all the students who participated last night, from our veterans to our rookies. For many of our younger students, it was their very first experience with participating in a high school math contest, and I know many of the questions were very hard, and sometimes the formats of all the different rounds got confusing, but I know that it will become more and more familiar over time. Thanks to all our high school students who welcomed our younger students onto their teams, especially our team captains: Andrew Ardito, David Bieber, Peixuan Guo, Brady Pelkey, Noah Rubin, and Felix Sun. You managed many challenges, including the fact that your teams were very heterogeneous, including various numbers of middle school students and other contest rookies. Several of the teams faced the handicap of being less than full strength (6 or 7 members), which made the relay rounds exceptionally challenging, especially the final one.

Special kudos to the very brave and courageous Guilderland and Shenendahoah teams, which were both under full strength in numbers AND composed primarily of middle school students. Although the relative inexperience of your teams last night posed many challenges, I know that both those teams are building a tremendous foundation for the future. Zubin Mukerjhee and Noah Rubin have been rising to the challenge by leading excellent middle school math circles in Guilderland. Felix Sun and Eric Wang have been coaching the MATHCOUNTS teams at the three Shen middle schools.

Kudos as well to Brady Pelkey from Hudson Falls, who captained a team otherwise composed of Niskayuna students. Brady is always someone we know we can count on to welcome new members to our math circle and help them feel comfortable with challenging problems.

Our top three teams were HEEG; a composite team composed of students from Shaker HS/JHS, Bethlehem HS, and Albany Academy; and Niskayuna Red, a team composed of students from Niskayuna High School, Iroquois Middle School, and Van Anterp Middle School.

Team high scorers were: Paul Rapoport (Composite 1), Preston Law (HEEG), Dave Bieber (Niskayuna Red), Bradley Johnson/Matt Walsh (Niskayuna Green), Isaac Malsky/William Wang (Guilderland), Felix Sun/Eric Wang (Shen).

Thanks again to everyone who made this contest possible: the problem creators, the parent volunteers, the omnipresent Mr. Babbitt (who organized all the paperwork, picked up the essential pizzas for our break, recruited outstanding parent and alumni volunteers, and constantly anticipated countless small but critical details like discovering that most of the electrical outlets near the projector system didn't work due to the ongoing construction and finding one that did.) Thanks also to Professor Rita Biswas, who was constantly thinking about how all this is going to translate to Albany Area Math Circle's hosting of the state tournament on April 10, 2010.

We will get one more opportunity to run NYSML Seasonal this year--in February--I know that Rita, Bill, and I learned a lot about what works and doesn't work. We won't have many alumni around then, so we will need even more parent volunteers to help us out, but I'm confident that will be an even better experience, because of all we've learned from running our first NYSML Seasonal.

And I know we can translate it all into a GREAT NYSML State meet on April 10, 2010.

And I know that last night's rookies who may have the opportunity to participate in other high school contests will be much better prepared as a result of last night's experiences. Thanks again to all who made it possible.

Thanks also to Prof. Chad Orzel of Union College, who gave a very engaging talk last night about teaching physics to dogs. It was a nice intermission between the two halves of the contest. Our students, alumni, advisors, and parent helpers all enjoyed it. He deserves yet another separate post of his own, which he'll get after his book lands under our Christmas tree next Friday. (Hmmm, hope nobody in my family is reading this?)

Here are the slides from an earlier version of the talk he gave last night:

Quantum Physics for Dogs: Many Worlds, Many Treats?

View more presentations from Chad Orzel.

In the spirit of his talk last night, I'll simply add this postscript:

We will post the number, names, and existence of individual perfect scorers (if any) after the extended contest administration window ends at midnight on January 1, 2010. Such individuals (if they exist) are not eligible for individual team high scorer awards, due to rules that avoid award duplication.

So, in other words, a wave function as to the possible existence and identity of perfect scorers will collapse at midnight on January 1, when we will release the news to the world.

IMPORTANT SERIOUS REMINDER: All students who took last night's contest are reminded that they should NOT discuss it with any students who might have yet to take the contest. That means: Don't discuss content of the problems, difficulty of the problems, or scores that you or anyone else got. And, especially, do NOT post any such information on the Internet until the extended contest administration window closes on January 1, 2010.

Quantum physics treats!

What do dog treats have to do with quantum physics? We'll find out this Friday as Union College physics Professor Chad Orzel will be speaking to math circle students about his new book, How to Teach Physics to Your Dog. The book is hot off the press and due to hit bookstores any day now, but those who are impatient can read the first chapter here. (I guess if you're REALLY impatient, you could create a wormhole and time-travel to a future when the book will be available, but if you know enough physics to create that wormhole, maybe you're ready to tackle teaching physics to an even lower lifeform than a dog, say a planaria? Ooh--I know, bad pun. Also, I gather Professor Orzel's dog actually thinks she's a higher lifeform than humans.) In any case, the title of Prof. Orzel's talk will be Quantum Physics for Dogs: Many Worlds Many Treats?

This video may give you some idea of what's in store:



We'll be listening to Prof. Orzel's talk during the middle of math circle, while eating our customary human treats (pizza!), but we also have some special mathematical treats in store Friday as well. Upstate New York ARML head coach George Reuter has concocted an excellent new high school math contest, NYSML Fall, which students will be working on from 6 p.m. to 7 p.m. (team round and mega round) and from 8 p.m. to 9 p.m. (individual rounds and relay rounds!)

To make it even cooler, it turns out that out of the three reviewers who vetted the problems for the new NYSML Seasonal contest (Mike Curry, Beth Schaffer, and Tom Zink), two are former captains of Albany Area Math Circle! And to top it all off, yet another former captain of Albany Area Math Circle, Drew Besse, will be home for the holidays in time help us out with scoring and proctoring the contest! If there are any other math circle alumni who will be home in time to join us Friday, please do let us know! We won't be meeting on December 25, but also welcome alumni to join us at our January 1 meeting to be held at RPI.

Congratulations to our AMC8 students

Scores of mathematically strong, enthusiastic, and courageous students from middle schools coached by Albany Area Math Circle student coaches and/or who are members of Albany Area Math Circle's middle school math circle took the AMC8 contest at one of the following three locations: Hebrew Academy of the Capital District, Iroquois Middle School, and Van Antwerp Middle School.

We would like to recognize the students who achieved national honors from the Mathematical Association of America's American Mathematics Competitions, the sponsor of the contest. Their names will be listed in a national honor summary book to be published next year.

Six Albany Area middle school students achieved National Honor Roll of Distinction scores. Their scores of 20 or above on this year's contest placed them among the top 1% of an already exceptional and self-selected pool of students nationwide who took the contest across the country.

Jien Ogawa Home Educators Enrichment Group (23)
Gili Rusak Shaker Junior High score (23)
Matt Gu Farnsworth Middle School score (22)
Aniket Tolpadi Iroquois Middle School (22)
William Wang Farnsworth Middle School (21)
Cecilia Holodak Van Antwerp Middle School (20)

Nine Albany Area students made the AMC8 Honor Roll with scores that placed them among the top 5% of an already extremely strong national pool of students taking the contest. This year that honor required a score of 17 to 19 points.

Isaac Malsky Farnsworth Middle School (19)
Chen Qu Acadia Middle School (18)
Alexander Wei Van Antwerp Middle School (18)
Zachary Benson Hebrew Academy of the Capital District (17)
Rajesh Bollapragada Van Antwerp Middle School (17)
Ziqing Dong Farnsworth Middle School (17)
Martin Schreiner Van Antwerp Middle School (17)
Isaac Smith Home Educators Enrichment Group (17)
Vineet Venangula Iroquois Middle School (17)

The students on the two honor lists above have been invited to join our high school students in taking the first NYSML Seasonal Contest this coming Friday.

We are proud of these students and ALL the local students who attempted this very difficult exam, whether or not their scores were honor roll. The contest was especially challenging this year. For some very strong and promising students, it was their very first math contest ever, an exciting chance to "get their feet wet," in a contest math experience. Regardless of scores and honors, Albany Area Math Circle is very proud of all who embraced the challenge this year.

For students who may not have done quite as well as they had hoped, please be assured that there are many more fun opportunities ahead, some of which will be open to all interested middle school students whose parents feel they are ready for more challenges. I will be talking about some of those opportunities at my talk this Thursday evening, December 17, at the Guilderland Public Library at 6:30 p.m.

We also applaud all the schools which offered students the opportunity to participate this year. Iroquois and Van Antwerp offered the AMC8 contest for the first time ever! Hebrew Academy has offered the contest for many years, and was especially gracious in hosting students from other schools which did not offer that opportunity to their students. We hope that even more schools around the Capital District will offer this challenging learning opportunity to their students in the future. Thanks to the principals, teachers, and parents who support those efforts this year, especially Alexandra Schmidt of Hebrew Academy who made outstanding arrangements for her school to host such a large group of math circle students.

We would also like to recognize the student coaches who helped to encourage and prepare students for this opportunity: Zubin Mukerjee and Noah Rubin have both been leading their own satellite middle school math circles in Guilderland serving students from a number of schools. Many of the students who took the AMC8 are also members of the middle school math circle that meets in Niskayuna, where high school students Andrew Ardito, Matthew Babbitt, Zubin Mukerjhee, Felix Sun, and Anagha Tolpadi have been mainstays of the fall coaching staff, taking on strong leadership roles, assisted by a number of other math circle veterans who have also been helping out this fall, including Preston Law, Schuyler Smith, Wyatt Smith, and Ved Tanadve. A number of our veterans have also been serving as the lead student coaches for local MATHCOUNTS teams: Matthew Babbitt for the homeschoolers, Anagha Tolpadi for Iroquois, Dave Bieber for Van Antwerp, and Felix Sun and Eric Wang for the three Shenendahoah middle schools.

Thanks also to adult AAMC volunteer advisors Bill Babbitt, Rita Biswas, and Alexandra Schmidt, as well as MATHCOUNTS parent coaches Kathryn Lesh and Anil Tolpadi for all their support and assistance with our middle school outreach efforts.

ARML Power Contest

Albany Area Math Circle's team is tied for 8th place among the 60-plus strong teams from around the country and the world that participated in the November ARML contest. ARML Power is a nice example of massively collaborative mathematics, as the rules allowed all two dozen students who attended that evening to participate! As the list of the top 30 teams below shows, we have some very distinguished company!

Three good guidelines from Timothy Gowers

These guidelines apply fully as well to collaborative problem-solving in math circles as they do to mathematical research.

3. When you do research, you are more likely to succeed if you try out lots of stupid ideas. Similarly, stupid comments are welcome here. (In the sense in which I am using “stupid”, it means something completely different from “unintelligent”. It just means not fully thought through.)

4. If you can see why somebody else’s comment is stupid, point it out in a polite way. And if someone points out that your comment is stupid, do not take offence: better to have had five stupid ideas than no ideas at all. And if somebody wrongly points out that your idea is stupid, it is even more important not to take offence: just explain gently why their dismissal of your idea is itself stupid.

5. Don’t actually use the word “stupid”, except perhaps of yourself.

Massively collaborative mathematics!

When I was a kid, I never imagined math as a collaborative activity. I imagined mathematicians as lonely hermits, working in isolation. It was only after I became an adult that I started to realize how completely wrong-headed those beliefs were.

Next Thursday evening, I'll be giving a talk to middle school math circle parents at the Guilderland Public Library at 6:30 p.m. about things they can do to support their students' development of problem-solving. One of the things I'll be talking about a good deal is the importance of mathematical communities and collaboration.

My favorite idea on the NYT annual "Year in Ideas" list is a great illustration of the power of collaboration in mathematics:

Massively Collaborative Mathematics

In January, Timothy Gowers, a professor of mathematics at Cambridge and a holder of the Fields Medal, math's highest honor, decided to see if the comment section of his blog could prove a theorem he could not.

In two blog posts — one titled "Is Massively Collaborative Mathematics Possible?" — he proposed an attack on a stubborn math problem called the Density Hales-Jewett Theorem. He encouraged the thousands of readers of his blog to jump in and start proving. Mathematics is a process of generating vast quantities of ideas and rejecting the majority that don't work; maybe, Gowers reasoned, the participation of so many people would speed the sifting.

The resulting comment thread spanned hundreds of thousands of words and drew in dozens of contributors, including Terry Tao, a fellow Fields Medalist, and Jason Dyer, a high-school teacher.

It makes fascinating, if forbiddingly technical, reading. Gowers's goals for the so-called Polymath Project were modest. "I will regard the experiment as a success," he wrote, "if it leads to anything that could count as genuine progress toward an understanding of the problem." Six weeks later, the theorem was proved. The plan is to submit the resulting paper to a top journal, attributed to one D.H.J. Polymath.

By now we're used to the idea that gigantic aggregates of human brains — especially when allowed to communicate nearly instantaneously via the Internet — can carry out fantastically difficult cognitive tasks, like writing an encyclopedia or mapping a social network. But some problems we still jealously guard as the province of individual beautiful minds: writing a novel, choosing a spouse, creating a new mathematical theorem. The Polymath experiment suggests this prejudice may need to be rethought. In the near future, we might talk not only about the wisdom of crowds but also of their genius. JORDAN ELLENBERG

Hat tip: Mrs. Ardito!

Adventures in creative problem-solving

George Mason University economist Alex Tabbarok describes an MIT team's very clever prize-winning problem solving to the DARPA Network challenge last week.

Earlier this week DARPA, the Defense Advanced Research Projects Agency, moored ten, 8 ft red, weather balloons in undisclosed locations across the United States.

The DARPA Network Challenge offered a prize of $40,000 to the person or group who first identified all the locations.

The MIT Group which won the challenge used a clever pyramid incentive scheme. Each balloon was worth $4000. The person to identify the location earned $2000. The person who invited that person to join the MIT group got $1000, the person who invited the person who invited the person who located the balloon got $500 and so forth (any money not distributed in this way was given to charity.)

The incentive scheme meant that contestants not only had an incentive to find balloons they had an incentive to find someone who could find balloons (or find someone who could find someone who could find balloons and so forth).

Incredibly, the MIT team located all ten balloons in just under 9 hours! The challenge may seem frivolous but in fact is a great example of how prizes and network technologies can combine to collect and use highly dispersed information--a problem of very general interest and relevance.

For those Albany Area Math Circle members working with middle school students, you can turn this news into an interesting math challenge. If you think about it, when the MIT team set up their prize-winning approach, they didn't know how long a chain of people it would take to find each balloon. It's possible that some of the chains could have hundreds or thousands of people in them. How did they know they'd have enough money to pay all the prizes they were offering?