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.

## Thursday, December 31, 2009

## Tuesday, December 29, 2009

### the generosity of math team captains

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

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

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.

## Monday, December 28, 2009

### 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.

## Sunday, December 27, 2009

### 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.

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.

## Tuesday, December 22, 2009

### 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

## Saturday, December 19, 2009

### 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:

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.

## Sunday, December 13, 2009

### 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.

## Saturday, December 12, 2009

### 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.

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.

## Friday, December 11, 2009

### 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:

Hat tip: Mrs. Ardito!

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!

## Thursday, December 10, 2009

### 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?

## Monday, November 30, 2009

### Eyes on the prize: the generosity of math coaches!

Professor Don Davis of Lehigh University has been doing a very impressive job of coaching the Lehigh Valley Math Team for many years. He founded the team in 1993 with a single team that placed 46th out of 73 teams. The team has improved over time, but it really took off in 2003, and there are now four teams Lehigh Fire, Lehigh Ice, Lehigh Thunder, and Lehigh Lightning. He now has two extremely impressive coaches assisting him with coaching these expanded teams: his former student Professor Ken Monks of the University of Scranton and MIT Senior Maria Monks, a former member of the Lehigh Fire team, which won the national championship in 2005.

The Lehigh Valley teams have had a string of extraordinary successes since 2003, but 2009 has been the most impressive ever. In February 2009, Lehigh Valley Fire began by winning first place among 154 teams in both the team-based collaborative rounds at Harvard-MIT Math Tournament. In June 2009, the Lehigh Valley team then went on to win its second national championship at ARML, this time by an astounding margin of 14 points over the nearest competitor, defending champion Phillips Exeter. In November 2009, the Lehigh Valley team capped off its year with yet another impressive first place finish in a very strong field at the Princeton University Math Tournament (PUMaC.)

How do they do it? If this were a football team or a basketball team or a hockey team, the keys to their amazing success would be a closely guarded secret. But they're a math team, and the culture of math team coaches has an incredible spirit of generosity and sharing, a tradition not found in athletics. There's a long tradition of math coaches generously sharing the materials, resources, and ideas they've developed for their teams with other coaches, and that tradition also spills over to the students as well. You can see that spirit when students from teams all over the country help one another out on Art of Problem Solving forums, for example. There is a belief that everyone benefits by encouraging and stimulating the mathematical growth of others, whether it's a teammate or a competitor from another team. That's because the ultimate prize is not a trophy or a medal or bragging rights, but rather the problem-solving skills, the camaraderie, the joy of community, the teamwork, the shared "Aha!" moments, and the esprit de corps that comes from working together on a challenging shared common goal.

So what have the Lehigh Valley math team coaches shared about the keys to their success? They've made it all an open book, and it's one well worth reading in full.

Professor Monks' general advice to students on preparing for competition is here, and it's an excellent list of books, practice problem, resources and ideas for preparation that I heartily endorse as well.

Professor and Maria Monks have also put together two extremely useful mathematical "play books," one playbook designed for MATHCOUNTS students and one playbook for high school contests. Albany Area Math Circle students should put both playbooks to good use. The MATHCOUNTS playbook is an excellent resource for those of you who are mentoring middle schoolers, and it's also the foundation upon which the high school playbook builds.

Some points Professor Monks makes are especially worth underscoring:

The Lehigh Valley math teams clearly have many extraordinary individuals, but their success is even more clearly far more than the sum of their individual talents. The Lehigh Valley students have clearly learned to work effectively together on the critical collaborative skills needed to grow as a team and to create a friendly and welcoming environment that has attracted students from a large geographic area, with some team members traveling as much as 80 miles to come to weekly practices.

And above all, it's clear that Lehigh Valley shares the same philosophy I've seen in all successful coaches: the more of your mathematical knowledge and understanding that you share with others, the more you will grow mathematically yourself.

The Lehigh Valley teams have had a string of extraordinary successes since 2003, but 2009 has been the most impressive ever. In February 2009, Lehigh Valley Fire began by winning first place among 154 teams in both the team-based collaborative rounds at Harvard-MIT Math Tournament. In June 2009, the Lehigh Valley team then went on to win its second national championship at ARML, this time by an astounding margin of 14 points over the nearest competitor, defending champion Phillips Exeter. In November 2009, the Lehigh Valley team capped off its year with yet another impressive first place finish in a very strong field at the Princeton University Math Tournament (PUMaC.)

How do they do it? If this were a football team or a basketball team or a hockey team, the keys to their amazing success would be a closely guarded secret. But they're a math team, and the culture of math team coaches has an incredible spirit of generosity and sharing, a tradition not found in athletics. There's a long tradition of math coaches generously sharing the materials, resources, and ideas they've developed for their teams with other coaches, and that tradition also spills over to the students as well. You can see that spirit when students from teams all over the country help one another out on Art of Problem Solving forums, for example. There is a belief that everyone benefits by encouraging and stimulating the mathematical growth of others, whether it's a teammate or a competitor from another team. That's because the ultimate prize is not a trophy or a medal or bragging rights, but rather the problem-solving skills, the camaraderie, the joy of community, the teamwork, the shared "Aha!" moments, and the esprit de corps that comes from working together on a challenging shared common goal.

So what have the Lehigh Valley math team coaches shared about the keys to their success? They've made it all an open book, and it's one well worth reading in full.

Professor Monks' general advice to students on preparing for competition is here, and it's an excellent list of books, practice problem, resources and ideas for preparation that I heartily endorse as well.

Professor and Maria Monks have also put together two extremely useful mathematical "play books," one playbook designed for MATHCOUNTS students and one playbook for high school contests. Albany Area Math Circle students should put both playbooks to good use. The MATHCOUNTS playbook is an excellent resource for those of you who are mentoring middle schoolers, and it's also the foundation upon which the high school playbook builds.

Some points Professor Monks makes are especially worth underscoring:

Perhaps the most valuable resource you have for preparing for ARML is ... each other! We have the best and friendliest mathematics students and coaches in our area on the team, all sharing a joy and passion for problem solving mathematics. If you have a question, ask a teammate!

...

Naturally, attending our ARML practices is very important in order to meet and interact with your teammates. and to practice the contest itself in a group setting. Here are some ways that you can reap the most benefit from our practices and do the most good for the team.

Cooperation: With 15 students trying to work together to solve problems on the Team and Power rounds, cooperation is essential to our success. Every decision you make during these rounds should be made with the good of the team in mind. There is no room for ego and bravado. How can I help? should be your mantra. Do something useful at all times. If you are not solving a problem, you can be writing a solution. If you are not writing a solution, you can be proofreading a solution. If you are not proofreading, you can be independently verifying an answer or solution. If you are stuck ask for help. If you can offer help to someone else, offer it. If you are good with a calculator you may be able to write a useful program to check something, if you are not, partner up with one of your teammates who is.

Respect: Everyone on our team is a superb mathematics student, usually the best math student in their school or local area. It is important to respect your teammates and their mathematical ability, both in terms of trusting their mathematical judgments and also from the ordinary aspect of being collegial to others. Be supportive of each other. Provide encouragement when someone is having a bad day (everyone has good days and bad days on math contests). Experienced ARML students should provide leadership and guidance for the newcomers. More advanced students can provide mathematical guidance to students with a weaker math background.

Communication: Talk to each other during practice. Make an effort to learn each other's names. It is difficult to cooperate in a group situation when you can't refer to each other in practice. We will provide you with name tags to facilitate this. During the Team and Power rounds, talk to each other within your squads, coordinate your activities, keep your team captain informed of essential information, share insights with the entire group. Between rounds and before and after practice get to know each other.

The Lehigh Valley math teams clearly have many extraordinary individuals, but their success is even more clearly far more than the sum of their individual talents. The Lehigh Valley students have clearly learned to work effectively together on the critical collaborative skills needed to grow as a team and to create a friendly and welcoming environment that has attracted students from a large geographic area, with some team members traveling as much as 80 miles to come to weekly practices.

And above all, it's clear that Lehigh Valley shares the same philosophy I've seen in all successful coaches: the more of your mathematical knowledge and understanding that you share with others, the more you will grow mathematically yourself.

### Inspiring stories

Everyone gets discouraged at times in life. Sometimes the strongest students get the most discouraged of all. Such students expect a lot from themselves, and it can be hard to live up to one's own high expectations. It's important to keep things in perspective. One good way to deal with such feelings when you are struggling is to find other people who can use your help. Everybody wins when you do that: it feels good to help someone else and explaining a tricky mathematical idea can deepen your own understanding of that idea.

The best way to learn is to teach somebody else. Math circle alumna Beth Schaffer is a great example of that idea in practice. Back when she was a Guilderland High School student coaching younger students, Beth wrote a couple of essays she has given me permission to share.

The first essay is one she wrote about her own middle school struggles, starting in sixth grade. She wrote it when she was a high school student in order to encourage the middle school team she was coaching at the time--that team went on to win the state championship by the biggest margin in recent history.

Beth's essay underscores a very important lesson--don't let yourself get discouraged by comparing yourself with other students--instead, keep on working on figuring out what you can, be willing to ask questions when you can't, and encourage and help everyone around you whenever you can. Not every team can win every contest, but every team can win something far more important than contests--the ultimate prize of developing their problem-solving skills, their ability to work collaboratively, their ability to communicate mathematically, and their ability to create a mathematical community that encourages others.

Her second essay, originally written for her college applications, described the spirit of Albany Area Math Circle meetings so well that I again asked for permission to share it, which she also granted. A few details of our meetings have changed since she wrote that essay--for example, we now meet on Fridays instead of Sundays, but her essay beautifully captures the essence of what our math circle is all about.

A willingness to work hard, to persevere when the problem seems impossible, a willingness to try half-baked ideas and to try simpler versions of the problem and to make mistakes and to celebrate and learn from those mistakes and to ask questions and share your half-baked ideas bravely without fear of looking foolish--and to encourage others to try their own half-baked ideas and to ask such questions--that is the heart of a student who will get the most out of participating in Albany Area Math Circle. That's also the heart of someone who will get a lot out of life.

Beth is now an MIT junior, where she has continued to work with younger students through MIT's Splash and ESP Programs as well as at Girls Angle. When she was a high school student, she participated in the Harvard-MIT Math Tournament (HMMT) as a member of Albany Area Math Circle's teams. Beth is now a tournament co-director of HMMT. She is pictured above with HMMT co-director, Winston Luo, on what must have been an intensely crazy day of running a large local contest earlier this month....somehow they are both smiling!

## Tuesday, November 24, 2009

## Sunday, November 22, 2009

### Prime number monopoly

By popular request from an awesome teacher friend, here are directions for Prime Number Monopoly, a fun game that I taught to teachers attending the New York City Math Teacher Circle retreat last summer. This is also a great game for older students to teach their younger siblings. Three students makes a good-size group for this game, but you can play with two or four per group as well.

There's minimal equipment involved. Each group needs a set of deeds (see below), a pair of dice, and paper and pencil for keeping track of scores. That's it. No playing board is needed. No play money is needed either, because students keep track of their balances as a running account.

Start by creating a set of deeds for the prime numbers for each group of students. You can probably get your students to figure out my pricing scheme from the picture of the first three deeds shown above. Once they figure out my scheme, they'll know what the price should be for larger primes such as 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, and 61. They can create create their own deeds using index cards and neatly lettering them.

I usually just use the primes from 2 to 11 in the games I run, and here is a nice PDF that creates two sets of professional-looking deeds for those numbers. The PDF is sized to print out cards on standard business card stock, which is pretty easy to get, or you can just print them out on regular paper, cut them out, and attach them to card stock for durability.

One student in each group needs to be the accountant, who will need to keep track of the running totals of money each student accumulates on a scoresheet, so the accountant should start by creating a column labeled with each student's name to track their winnings. Other students need to regularly "audit" his work by watching him to make sure he gets everything recorded properly. (Don't want any budding Bernard Madoffs!) You may want to rotate the job of accountant. Alternatively, each player can keep track of his own earnings--again audited by the other players!

At the beginning of the game, nobody has any money, so there should be a zero at the top of each column. Also, at the beginning of the game, nobody owns any prime numbers, so they should all be placed at the center of the table.

Each student takes turns rolling the two dice. Say he rolls a 5 and 4. Then, he has a choice of declaring his total winnings for that round to be $54 or $45. (You are probably wondering why anyone would choose to go for $45, but you'll see that will sometimes be a useful strategy in later rounds.) Whichever amount he chooses is the amount the accountant records in the books as his winnings for that round. He should keep a running tally, so players know how much money they have.

At the beginning of any turn, BEFORE he rolls the dice, if a player has accumulated enough money to buy a prime number that is in the center of the table, then the player may state, "I'd like to buy a prime," and specifies the one he wants to buy. At that point, the accountant subtracts the appropriate amount of money from his total, and the player takes control of that prime, moving it to his place to indicate his ownership. He then rolls the dice as usual on his turn, and records any winnings as usual. From then on, however, when any other player rolls and declares a multiple of his prime, the other player has to SPLIT his winnings with the owner of the prime.

Example:

Jane has bought the deed for 7. John later rolls a 4 and a 2 on his turn, so he has the choice to declare $42 or $24. If John declares "$42," then John has to split his earnings with Jane because 7 is a factor of 42. Each will get $21 recorded in their columns. On the other hand, if John declares "$24," then he doesn't have to split it--he'll keep the whole $24.

More complicated example:

Suppose Jane owns the deed for 2 and John owns the deed for 11, and Jimmy rolls a 4 and a 4. Then Jimmy has no choice but to declare 44. Since both 2 and 11 are factors of 44, then it's a three way split. Each player will get 1/3 of the $44. (Generally, I run this game by having the students round off to the nearest whole number, which results in $15 per student in this case, because it encourages fast mental arithmetic, but it's your decision how to run the game with your students.)

Even more complicated example:

Same facts as above except that assume that Jane is the one to roll the 44. She owns the 2 and John owns the 11. In that case, the split is 1/3 to John and 2/3 to herself.

I generally set a predetermined amount of time for the game to run. At the end of the game, all students can sell their deeds back to the bank for 90% of their original value (good opportunity to talk about asset depreciation!) and the student who has accumulated the most money wins.

After the students have played a game, it's a good idea to discuss strategies. Are some prime numbers better bargains than others? Should you buy the cheapest prime you can afford as soon as you have the money to do so, or should you save up your money and buy a somewhat more upscale prime? This question turns out to be a bit more complicated than people might initially think. Two seems like it would be a great deal (because half of all numbers are divisible by 2), but 3 has special advantages not available to 2.

Some of the same sorts of considerations come up in the game of regular Monopoly. The value of a property depends in part on the amount of rent you can collect when someone lands on your property, but it also depends on the probability that someone will land there. It does little good to own a high rent property if people rarely land there! It turns out that someone named Philip Orbanes has exhaustively analyzed the probabilities of landing on all the different spaces on a Monopoly board. He wrote up his analysis in a great little book called The Monopoly Companion. The book is out of print now, but used copies are available very inexpensively.

Source: This game is loosely adapted from a game called "Pseudo-Monopoly," which I found in the book Family Math years ago, which also had other features, like an income tax, and you can of course adapt this game to incorporate additional features of your choosing as well.

## Saturday, November 21, 2009

### Congratulations to our PUMaC Team

Eight Albany Area Math Circle students traveled to Princeton University this weekend for the fourth annual Princeton University Math Contest (PUMaC). AAMC students shown above relaxing before the very challenging contest are, from left to right: Wyatt Smith, Schuyler Smith, Jay White, Andrew Ardito (captain), Gurtej Kanwar, Zubin Mukerjee, Dave Bieber, and Matthew Babbitt. Also, in the background, you can see some of Albany Area Math Circle's outstanding parent volunteers. Left to right, they are Dr. Susan Bieber, Professor Rita Biswas, and Mrs. Eileen Ardito. The photo was taken by AAMC advisor Mr. Bill Babbitt. Their energetic efforts and enthusiastic support have made major contributions to our math circle.

AAMC students once again brought home a number of honors, including EIGHTH place overall team and FOURTH Place in the Power Round! The field at PUMaC has grown bigger and more competitive each year. It now attracts exceptionally strong teams from all over the country and the world. This year's field of 70 teams included two all-star teams from Beijing, as well as longtime traditional powerhouse teams from New York City, Phillips Exeter, Thomas Jefferson (VA), and many other places around the country and the world. Congratulations also to all the competitors, with special kudos to Lehigh Valley Fire (the reigning two-time national championship team from ARML), which took first place overall team honors. In fact, it's amazing to note that every national championship ARML team from 2002 through 2009 participated in PUMaC this year! What an honor for Albany Area Math Circle to do so well with such extraordinary competitors.

AAMC students brought home individual honors as well: Matthew Babbitt and Schuyler Smith tied for 9th place in the Number Theory competition. Andrew Ardito took 5th place in Number Theory. In the Combinatorics competition, Dave Bieber took 9th place and Andrew Ardito took the medal for second place, only one problem away from first place. As team captain, Andrew led the outstanding Power Round collaboration, with major contributions to the Power Round also coming from two other key veterans: Matt Babbitt and Dave Bieber.

The contest is entirely run by Princeton students. Princeton undergrads manage promotion, registration, finances, get sponsorship, create the problems, score and record results, assist distant teams with travel arrangements, maintain the website, arrange for catering lunch, run awesome minievents, and much more. The Princeton undergraduate math students who ran the contest deserve great praise for all their attention to organizational details, which made this year's contest the smoothest run ever.

Albany Area Math Circle alumna Alison Miller, who is now a first year grad student at Princeton, volunteered as a grader at the contest. (She recused herself from scoring Albany Area Math Circle papers.)

## Wednesday, November 18, 2009

### Math Prize for Girls contestants

Ashley Cho, Heidi Chen, and Anagha Tolpadi at Saturday's Math Prize for Girls in Manhattan. We are proud to claim all three as members of the Albany Area Math Circle.

Ashley and Heidi attend Emma Willard School, where they are leading other students in preparing for the upcoming Harvard-MIT Math Tournament in February. Both are boarding school students from overseas; Ashley is a junior and Heidi is a senior. Anagha is a senior at Niskayuna High School.

Ashley is a 2008 USAMO qualifier who was a member of Albany Area Math Circle's NYSML team, which ranked 3rd in the New York State A-division in 2009. Ashley tied for 25th place at Saturday's contest, bringing home one of the beautiful crystal honorable mention trophies you can see sparking in the background of the photo at right. The competitive field was exceptionally strong, including many students from all over the country who have won medals in international competition, making Ashley's honor all the more impressive. Ashley was the highest scoring student from any school in New York State.

Heidi is also an outstanding math student, who was selected to participate in the 2009 Princeton Summer Workshop in Mathematics. Heidi has twice been named to the Upstate NY ARML team, an all-star math team representing all of Upstate New York at the national American Regions Math League (ARML) tournament held simultaneously at four college campuses across the country in June each year.

Anagha is a multi-talented student at Niskayuna High School, where she is editor-in-chief of the school newspaper. She got her start in math contests as a sixth grader at Iroquois Middle School, where her first math-team coach was AAMC member Alison Miller. She went on to achieve many honors, including qualifying for the American Invitational Math Exam while still an eighth grader. Now Anagha is paying it forward by coaching the Iroquois Middle School team herself. Two years ago, she coached the team all the way to the state finals, and she is working hard to encourage them again. She is also an enthusiastic mentor for our middle school math circle. Here's what she has written about her experiences working with younger students:

I volunteer my time coaching MATHCOUNTS at my former middle school. I had participated when I was in middle school and I had loved it then. As a coach, everything is certainly different, but honestly, I love it even more now. I expect a great deal from the kids, and they know it, which makes them work harder. To be able to help kids problem solve and teach them new concepts is like nothing else in this world. There is nothing better than watching a kid's face light up when he figures out a problem, knowing that you helped him get there.

All three of these outstanding students have clearly learned and put into practice the most important lesson we try to share with all members of Albany Area Math Circle: the best way to deepen your own understanding of mathematics and problem solving is to reach out and help someone else!

### Pictures at a competition

Thanks to Hebrew Academy of Albany for their gracious hospitality in hosting Albany Area Math Circle middle school students for the AMC8 math contest. We are grateful to Head of School Rabbi Rami Strosberg for the school's kind invitation and to math teacher Alexandra Schmidt for making all the arrangements and proctoring the contest.

Public school students from the Albany, Guilderland, North Colonie, and Saratoga Springs school districts as well as homeschooled students joined Hebrew Academy students for this exciting mathematical challenge, building new bridges of mathematical friendship in talking the problems over afterwards with milk and cookies.

Thanks also to all the math teachers, MATHCOUNTS coaches, and math circle mentors who have helped to prepare students to embrace and enjoy such challenges.

We are also grateful for the assistance of Iroquois Middle School Principal Victoria Wyld and Van Antwerp Middle School Principal Luke Racoczy, whose support made it possible for math circle students attending those schools to take the contest at their schools. We hope that more principals across the region will follow their leadership in the future, so that students everywhere will have a convenient and accessible opportunities to participate in this world-class challenge.

We are also grateful for the assistance of Iroquois Middle School Principal Victoria Wyld and Van Antwerp Middle School Principal Luke Racoczy, whose support made it possible for math circle students attending those schools to take the contest at their schools. We hope that more principals across the region will follow their leadership in the future, so that students everywhere will have a convenient and accessible opportunities to participate in this world-class challenge.

IMPORTANT NOTE: Students all over the country--indeed all over the world!--take this challenging contest. A few schools with special circumstances are allowed to take the contest on a delayed date. Wherever you took your contest, please remember your proctor's instructions that you must NOT discuss the contest with anyone outside your immediate testing group until after November 24.

## Monday, November 16, 2009

### Math Prize for Girls problems now available to the world

As I mentioned in my post yesterday, the Math Prize for Girls problems are a terrific learning opportunity for everyone, regardless of gender or whether they were able to travel to participate in Saturday's contest. The problems are now available for download. (Use this URL if the embedded link doesn't work: http://mathprize.atfoundation.org/MathPrize2009problems.pdf)

If you want a place to discuss your solutions and alternative approaches, this Art of Problem Solving dedicated forum is the place to go. The problems' author, Ravi Boppana, is an active discussion participant on that forum, and mathematically passionate students from all over the world contribute their ideas.

I plan to work on these problems myself as I find time to do so. I can think of a few public officials who might also find working on some of these problems a valuable learning opportunity as well. For starters, there's US Education Secretary Arne Duncan and NYC School Chancellor Joel Klein. It would be good for them to see the kinds of challenging problems that these high school students attacked with such energy--on a Saturday morning! Maybe Mayor Bloomberg too! Then there's US Treasury Secretary Tim Geithner (and, no, Mr. Secretary, you may NOT use TurboTax--or even a calculator--on these problems!) And, of course (mischievious grin!), maybe White House Economic Advisor Larry Summers would enjoy working on these math problems, when he needs a break from solving the country's economic problems.

Imagine the possibilities for Intrade betting if those folks had been "celebrity contestants" allowed to try the problems under timed and proctored conditions. My money would have been on quite a few of those high school girls to beat all the government officials listed above.

## Sunday, November 15, 2009

### Math Prize for Girls highlights

A century ago, suffragettes gathered in Cooper Union's Great Hall, agitating for the right to vote for women. I believe they would have been bursting with pride if they could have seen the first annual Math Prize for Girls awards ceremony held in that hallowed and historic hall yesterday afternoon.

The winner was

**Elizabeth Synge**from Massachusetts, taking home a prize check for $20,000 for top honors on a very challenging set of problems during yesterday's written test held at NYU's Courant Institute of Mathematics. This prize was yet another impressive accomplishment for Elizabeth, shown above in a photo taken last summer at an international competition in China, where she also won high honors. You can read more about Elizabeth's background here.**Joy Zheng**of Washington,

**Lynnelle Ye**of California, and

**Jane Wang**of New Jersey all tied for second place in the contest and took home checks for $6,000 each.

Albany Area Math Circle's own

It was an honor just to qualify for an invitation to such an event, and Albany Area Math Circle was also proud to be represented by Ashley's schoolmate

**Ashley Cho**from Emma Willard School, who tied for 25th place, brought home a beautiful sparkling crystal engraved trophy for Honorable Mention. Complete results for the top 31 students are listed here. All 220 students took home Mathematica software donated by Wolfram as well as new friendships forged with kindred spirits talking over the problems after the competition.It was an honor just to qualify for an invitation to such an event, and Albany Area Math Circle was also proud to be represented by Ashley's schoolmate

**Heidi Chen**from Emma Willard as well as**Anagha Tolpadi**from Niskayuna High School. Younger Albany Area Math Circle students have already been inspired to work hard in hopes of qualifying in future years.As advisor to the Albany Area Math Circle, I have attended scores of math contests but never have I attended one quite as unusual--or as visually spectacular--or as well-organized!--as this one.

I have some brief notes and observations below--more will follow when I have time.

#1) To my knowledge, never have so many young women with so much mathematical promise come together in a single place before. The gathering provided an opportunity for a girl who might have few kindred spirits sharing her passion in a small hometown to make friends and form bonds with a community of young women sharing her interests from all over the country. The students attending the event have been exchanging email addresses, cell phone numbers, and Facebook friend invitations.

To give some idea of the strength of the field at the Math Prize for Girls, the 220 contestants from around the country included numerous USA Math Olympiad qualifiers as well as members of recent US China Girls Math Olympiad (CGMO) teams who are still in high school, including gold, silver, and bronze medalists from that prestigious international competition.

To give some idea of the strength of the field at the Math Prize for Girls, the 220 contestants from around the country included numerous USA Math Olympiad qualifiers as well as members of recent US China Girls Math Olympiad (CGMO) teams who are still in high school, including gold, silver, and bronze medalists from that prestigious international competition.

#2) There were 20 excellent and very challenging problems for the girls to tackle during the 2.5 hour morning written exam, mostly created by the contest's extraordinary director, the highly talented Ravi Boppana. He was assisted by a stellar advisory board, including mathematicians from Berkeley, Harvey Mudd, MIT, Princeton, and Stanford. I hope the Math Prize for Girls will publish their problems on the website, because they will be a great problem-solving resource and learning opportunity* for all students to work on, no matter what their gender, where they live, or whether they were able to take part in person.

#3) While students were taking the written test, Richard Rusczyk, from Art of Problem Solving, gave an outstanding talk to parents with excellent and eminently sensible advice about opportunities and resources for encouraging students to develop their problem-solving abilities. I agreed with pretty much everything he said, except there were things I wanted to add and underscore. After the slides from his talk are posted to the contest website, I'll create a post adding my own thoughts.

#4) Mathematician Cathy O'Neil gave a superb keynote speech, which struck exactly the perfect chord. It's not easy to give a talk that engages a large group of high school students you've never met before who are mathematically exhausted after a morning of working on very hard math problems and who are nervous and excited about the upcoming announcement of awards and a possible tie-break round, but she delivered a sparkling talk on a fascinating topic she'd first encountered on an old HCSSiM "Interesting Test." Her discussion focused on an application of directed graph theory problems known as "chicken pecking order" problems. There are many related explorations math circle students can explore, and you can find some good related problems in Sam Vandervelde's list here.

#5) The other shorter talks were great as well. I enjoyed them all. I have sat through scores of tedious awards ceremony speeches over the past ten years--these were the best ever.

#6) The newly renovated historic Cooper Union Great Hall provided an awe-inspiring and highly appropriate setting for the awards ceremony. Presidents from Lincoln through Obama have spoken here. Many important social movements, including women's suffrage, took root here. What a terrific place to promote a new movement to encourage even more young women across the country to embrace mathematical challenges.

#7) There was a superb team of volunteers assisting with the contest, including many familiar faces and names I knew from the New York City Math Teachers' Circle or NYSML. Volunteers I recognized included Deanna Abramowitz, Tim Evans, Maggie Feurtado, Sheila Krilov, Marie Parham, Ming Jack Po, Amy Prager, and others whose faces were familiar but whose names escape me. Many of the volunteers were highly dedicated New York City math teachers, going above and beyond their usual duties after a hard week of work in the classroom.

Thanks to so much able and highly experienced help, proctoring and scoring went extremely well. I was sitting very close to the side of the stage and found myself extremely impressed by the exceptionally smooth and graceful execution of the simultaneous and complex tie-breaking round to break ties for second, fifth, and eighth place. It's volunteers like these who make such contests possible.

#8) One highly talented young woman,

**Meena Boppana**, was present at the event, but notably absent from the competition. Meena is a Hunter College High School sophomore and already has established an outstanding record as a New York mathematical rock-star. Since her dad was the director of the Math Prize for Girls event, she was of course not eligible to compete, but she and her mother, Dr. Ranu Boppana, were both enthusiastic and very helpful volunteers.As a seventh grader, Meena made mathematical history by winning the Manhattan Chapter MATHCOUNTS written contest, almost surely the first girl to do so. As an eighth grader, Meena ranked 2nd in the statewide MATHCOUNTS competition and represented the great state of New York as a member of the state team at Nationals in Denver. As a ninth grade freshman, Meena made the citywide A-division high school team which won the New York State Math League championship.

#9) I felt an overwhelming sense of "grandmotherly" maternal pride at this event. I don't have any actual biological grandchildren so you might be wondering how this could be possible.

So many of the participants were students that my daughter Alison had mentored through her work as a mentor or coach or instructor in programs such as China Girls Math Olympiad, MathCamp, Girls' Angle, MATHCOUNTS, and the Math Olympiad Summer Program, that I realized that I was in some sense a "mathematical grandmother" of many of these girls. In fact, some of the girls may even have been been mentored and coached by still other students whom Alison had mentored, which makes me their "mathematical great-grandmother" (or who knows, maybe even a mathematical great-great-grandmother....)

It makes me feel a little old....but I don't mind! And I also didn't mind the long lines in the women's restrooms at the end....it was nice to see all the networking among the girls going on in the lines (and waiting in the long lines gave me a chance to dispense a few more tips to other mothers interested in starting up math circles in their hometowns.) Women's restrooms are usually relatively deserted places during typical math contests--it was nice to be in a busy one for a change.

*In my opinion, the learning opportunity is a great one for adults as well as students. I plan to work on the contest problems myself. I also have a few ideas for some who could learn a lot from working on these problems, but I'll reserve those for a separate post.

NOTE: Updated 11/18/2009--thanks to Contest Director Ravi Boppana for providing information to set the record straight and correct a few errors in the earlier version of my post. The most significant error is that his daughter Meena actually attends Hunter College High School, rather than Stuyvesant as I had previously stated incorrectly.

### AMC8 coming Tuesday!

Good luck to students from Albany Area Math Circle who will be joining middle school students from all over the country--indeed all over the world!--in taking the AMC8 this Tuesday. Back when I was an 8th grader, I never could have imagined that math could be this much fun!

Here's one of my favorite problems from an old AMC8 (#19 2000). (Click on the image to make it larger and clearer.)

There are some tedious and time-consuming ways to solve this problem, but if you think about it for a while, there's a very beautiful and cool "outside the box" way to solve this problem very quickly and elegantly. (Or maybe we should call it a "redefine the box" way to solve the problem!) What is great about the cool "redefine the box" solution is that once you see it, it's very easy to check your work and be sure your answer is correct. (If you do it the tedious, hard way, there are more places to make silly mistakes and go astray.)

For hints and solutions to this and other old AMC8 problems, check out this link.

No matter whether you are taking the AMC8 with Albany Area Math Circle along with students at Hebrew Academy or at your own school (shout-outs and kudos to Iroquois Middle Schools and Van Antwerp Middle Schools, who will be administering the AMC8 for the first time this year!), you will be taking part in a great event along with almost 200,000 strong and enthusiastic problem-solvers around the world!

A few things to remember:

**What to bring:**a couple of #2 pencils with erasers. (Good mathematicians make mistakes, so good erasers are important!)

**What NOT to bring:**Please do NOT bring any calculators. Their use is NOT permitted under current rules.

**What to expect:**Challenging outside-the-box math problems designed to encourage middle school students to stretch themselves mathematically. They are harder than regular classroom math tests because they require insight and creativity, but they are also more fun and rewarding to solve. There will be awards and recognition for high-scoring students, but the real prize is the encouragement to children to learn and grow mathematically. Especially if this is your child's first experience with contest math, it's important to encourage realistic goal-setting.

Shooting for 100% is not realistic--in some years, nobody in the whole state gets all 25 questions correct, even though the contest is taken by thousands of strong math students across the state. About 170,000 strong math students take the contest nationwide, and there are even some international contestants from places such as China, Greece, Romania, and Turkey. Getting more than 50% correct is a strong above-average performance, even for an experienced 8th grader.

**My advice to pass on to your students:**I'll tell you what I told my own daughters when they took such exams. My motto is "Have a good time!" "Do the best you can and enjoy the problem-solving experience. As long as you enjoy learning from working on mathematical challenges, the experience is a successful one worth celebrating. I am proud of you just for embracing this challenge and struggling with these problems."

**After the contest:**One of the most valuable parts of the contest is the camaraderie and fun of talking over answers afterwards. There is time specifically set aside in the schedule to do this. For some problems, there are several different creative ways to approach and solve the problem, and it's great for students to compare and discuss those approaches afterwards.

**One important note:**It's fine for students to talk the problems over among your local testing group after the contest BUT you must NOT discuss the contest via email, Internet, or any other method with students outside that group until after November 24. Some schools receive special permission to take the contest late for a variety of reasons, and it is important not to spoil the integrity or fun of their contest experience by disclosing anything about the problems until that date.

## Saturday, November 7, 2009

### AMC8 Tuesday Nov 17

The AMC8 is coming up on Tuesday November 17. Some area middle schools will be offering this contest to their students. If you are interested in taking the contest, please make sure to check with your MATHCOUNTS coach and/or math teacher to see if your school is offering you the contest.

For those middle school math circle members attending schools that do NOT offer the contest, Albany Area Math Circle has arranged a special administration of the AMC8 to be hosted at Hebrew Academy of Albany. Please use this form to register your student. [

**Update**: registrations are

**now**closed.]

Albany Area math Circle is grateful to Hebrew Academy of Albany for its hospitality!

## Tuesday, October 27, 2009

### Great short math videos

Many of our math circle students are mentoring or coaching younger students. Here are some excellent model examples of presentations to inspire you.

Ina Petkova (Columbia math PhD student) gave a very elegant, simple and clear exposition of a proof of the Pythagorean theorem that middle school students should readily be able to understand and later to recreate for themselves. The Girls' Angle program in Cambridge Massachusetts created the video of her presentation as the first video in their series. I look forward to seeing more on their website in the future.

Richard Rusczyk has many excellent video presentations which you can watch on the Alcumus section of the Art of Problem Solving website, but you will need to solve some problems first in order to earn the right to see them. What I like about Richard's presentations is that he doesn't just demonstrate the "nice" way to do a particular problem, he also shows some of the not-so-nice ways we might initially think about trying. It's important to acknowledge to students that we don't always know at the outset exactly what approach is going to work best for a given problem. There's a certain amount of groping about that is often necessary to get insights into a good method to solve a problem. We need to demonstrate that problem solving is a process, not just a magic bag of tricks.

Ina Petkova (Columbia math PhD student) gave a very elegant, simple and clear exposition of a proof of the Pythagorean theorem that middle school students should readily be able to understand and later to recreate for themselves. The Girls' Angle program in Cambridge Massachusetts created the video of her presentation as the first video in their series. I look forward to seeing more on their website in the future.

Richard Rusczyk has many excellent video presentations which you can watch on the Alcumus section of the Art of Problem Solving website, but you will need to solve some problems first in order to earn the right to see them. What I like about Richard's presentations is that he doesn't just demonstrate the "nice" way to do a particular problem, he also shows some of the not-so-nice ways we might initially think about trying. It's important to acknowledge to students that we don't always know at the outset exactly what approach is going to work best for a given problem. There's a certain amount of groping about that is often necessary to get insights into a good method to solve a problem. We need to demonstrate that problem solving is a process, not just a magic bag of tricks.

## Wednesday, October 21, 2009

### A puzzle celebrating Martin Gardner's 95th birthday

John Tierney of the New York Times begins his feature article on the prolific and always delightful recreational mathematician Martin Gardner with a puzzle:

For the answer to the puzzle and the fascinating story of a mathematical legend, see For Decades, Puzzling People With Mathematics in today's New York Times.

You can also find more great Gardner puzzles on John Tierney's NYT blog here.

As the article notes:

Happy Birthday indeed! And many happy returns of the day!

For today’s mathematical puzzle, assume that in the year 1956 there was a children’s magazine in New York named after a giant egg, Humpty Dumpty, who purportedly served as its chief editor.

Mr. Dumpty was assisted by a human editor named Martin Gardner, who prepared “activity features” and wrote a monthly short story about the adventures of the child egg, Humpty Dumpty Jr. Another duty of Mr. Gardner’s was to write a monthly poem of moral advice from Humpty Sr. to Humpty Jr.

At that point, Mr. Gardner was 42 and had never taken a math course beyond high school. He had struggled with calculus and considered himself poor at solving basic mathematical puzzles, let alone creating them. But when the publisher of Scientific American asked him if there might be enough material for a monthly column on “recreational mathematics,” a term that sounded even more oxymoronic in 1956 than it does today, Mr. Gardner took a gamble.

He quit his job with Humpty Dumpty.

On Wednesday, Mr. Gardner will celebrate his 95th birthday with the publication of another book — his second book of essays and mathematical puzzles to be published just this year. With more than 70 books to his name, he is the world’s best-known recreational mathematician, and has probably introduced more people to the joys of math than anyone in history.

How is this possible?

For the answer to the puzzle and the fascinating story of a mathematical legend, see For Decades, Puzzling People With Mathematics in today's New York Times.

You can also find more great Gardner puzzles on John Tierney's NYT blog here.

As the article notes:

“Many have tried to emulate him; no one has succeeded,” says Ronald Graham, a mathematician at the University of California, San Diego. “Martin has turned thousands of children into mathematicians, and thousands of mathematicians into children.”

Happy Birthday indeed! And many happy returns of the day!

## Saturday, October 10, 2009

### Congratulations on AMC12 Team Standings!

The photo above shows Prof. Moorthy celebrating with the Albany Area Math Circle team of Dave Bieber, Yipu Wang, and Andrew Ardito. Their outstanding team score of 387 on the AMC12B was second place in New York State, and nipped at the heels of first-place powerhouse Stuyvesant (391.5) closer than ever before!

The photo above shows Albany Area Math Circle team of Jay White, Matthew Babbitt, and Andrew Ardito celebrating their excellent third place in New York performance on the AMC12A date.

Here are the top team score standings for New York State on both dates:

AAMC students who took the A-date contest at their schools also contributed to Merit Roll honors for team scores of 300 or above at Bethlehem High School, Emma Willard School, Guilderland High School, and Niskayuna High School.

Congratulations to all the schools listed above, and to all the hundreds of excellent high schools all over the state and thousands of schools across the country that offered the AMC contests to their students! If your school doesn't offer the AMC contests, check out this link and then talk to your math teacher!

The complete list of AMC honors for the 2009 AMC10 and AMC12 contests are available in the PDFs linked below. Students working on college applications who wish to look up their honor roll scores for the past six years can search for their names and/or school names in these PDFs.

2009 AMC10/12 Honors and Award Summary Book

2008 AMC10/12 Honors and Award Summary Book

2007 AMC10/12 Honors and Award Summary Book

2006 AMC10/12 Honors and Award Summary Book

2005 AMC10/12 Honors and Award Summary Book

2004 AMC10/12 Honors and Award Summary Book

## Friday, October 9, 2009

### Hard Problems: The Road to the World's Toughest Math Competition

Public television stations across the country will be showing Hard Problems, the documentary about the six American students who represented the United States at the 2006 International Math Olympiad. In the Albany area, WMHT (channel 17) will be showing movie on Sunday October 18 at 6 p.m. More information about the movie, including air times in other cities, is available here. Here's the movie's trailer:

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