## Tuesday, December 21, 2010

### Happy Holidays!

I have received a few holiday gifts I'd like to share with you:

1) A beautiful video from Ken Fan at Girls Angle

The program Ken used to make the video above is a free open source 3D creation suite you can use too. Available at www.blender.org

2) A pointer from Melissa Smith of the Ithaca Math Circle to a great article by Nate and Sam Cornell. Nate is a psychologist at Williams College and Sam is a writer in San Francisco. Their article really accorded with decades of experience for me as a lifelong educator who continues to love her own learning challenges.

Here is a link to the article:
A Really Hard Test Really Helps Learning: Challenging tests and falling short may be hard on the ego, but they can do more than mere studying for eventually getting it right

Here's an excerpt that especially struck Melissa and resonated well with me also:

Both studies independently indicate a striking fact. We tend to assume that the best way to consume and remember information is through the application of rigorous, extended study. What we fail to see, however, is that the process of trying to work through a problem to which we don’t know the answer focuses our attention on it in a way that simply studying it does not. The desire to get the answer right, and the frustration of failure, is partly to account.

But there’s another element as well. When we struggle to learn something, and fail, the moment we finally get the answer it imprints itself more deeply on our mind than it would have had struggle and failure not preceded it.

3) Excellent advice from mathematician Lillian Pierce, currently at the Institute for Advanced Study: Try again! Fail again! Fail better!

4) A random factaroony from Number Gossip: 2011 is the first odious prime number we've had since 1999. For more about "odious" and "evil" numbers, see here.

5) And one last present (especially nice during those bleak and gloomy times when it feels like winter may never end ... or when it feels that a problem may never get solved.) A hat tip to the Metroplex Math Circle for calling my attention to this enchanting video. Double-click on the video for a bigger view to see the whole picture.

For more about the mathematics behind this video, see this website.

And remember ... just as sometimes the most unpromising-looking larvae can turn into the most exquisite and magical dragonflies, so too can the most hopeless looking math problems turn into the most awesome aha! experiences.

May 2011 be filled with many wonderful problems and discoveries! And many wonderful mathematical friends with whom to share them!

## Tuesday, December 14, 2010

### AMC8 Congratulations!

Congratulations to all the Albany Area middle school students who embraced the challenge of the AMC8 with such enthusiasm this year! Congratulations as well to our high school students who have mentored many of them.

Honor Roll (scores of 18 to 21)

18 Alex Cao Shaker JHS
18 Alex Wei Van Antwerp MS
20 Patrick Chi Iroquois MS
20 Gideon Schmidt Iroquois MS
21 Shreya Arora Iroquois MS
21 Andrei Akhmetov Van Antwerp MS
21 Alicia Chen Farnsworth MS
21 Zach Benson Hebrew Academy of the Capital District
21 Sean Setzen Hebrew Academy of the Capital District

Honor Roll of Distinction (scores of 22 to 25)

23 Gili Rusak Shaker JHS
24 William Wang Farnsworth MS
25 Ziqing (Bill) Dong Farnsworth MS

Bill Dong's perfect paper of 25 is extremely unusual--there have been many years when there are no 25's anywhere in the entire state. It is all too easy for even the strongest students to make mistakes on a test like the AMC8, as many of our veteran high school students and alumni can tell you!

Bill joins a very rarefied group of students--Albany Area Math Circle's only previous perfect scorers on the AMC8 have been Raju Krishnamoorthy (1999), Drew Besse (2001), Schuyler Smith (2006), and Matthew Babbitt (2007).

Bill will also receive the AMC8 high scorer in the state award, as did Raju, Drew, Schuyler, and Matthew, as well as Andrew Ardito and Dave Bieber.

Students who would like to prepare for next year can find lots of old AMC8, AMC10, and AMC12 test problems, along with hints and solutions free and online at this webpage.

### Fall Middle School Math Meets

Thanks to everyone who contributed to the success of our inaugural season of Fall Math Meets! Watching you smile and make new mathematical friends and share cool math as you talked over the problems together after the contest is what makes it all worthwhile for us volunteer advisors.

There are many ways to measure success--we recognize a few of them here, but the ultimate measure of success is whether you continued thinking about and talking over the new ideas you learned afterwards.

October High Scorers:

8 points:
Zachary Benson, Sophie Rich, Helen Yuan

9 points:
Emily Honen, Jeffrey Shen, Cathy Shi, Max Thomas

10 points:
Bill Dong, Thomas Glozman, Ben Salem, Gideon Schmidt, Philip Sun, William Wang

"Whole is more than the sum of the parts" team award:

November High Scorers:

6 points:
Zachary Benson, Gwenda Law, Michelle Yu

9 points:
Alex Cao, Patrick Chi, Jerry Qu, Gideon Schmidt, William Wang

10 points:
Alicia Chen, Bill Dong, Philip Sun, Alex Wei

High scoring newcomer award:
Samuel Enriquez, Rafi Nizam, Jason Tang, Michael Zhu

"Whole is more than the sum of the parts" team award:
Jason, Mike, Zach

December High Scorers:

7 points:
Alex Cao, Thomas Glozman, Michelle Yu

8 points:
Bill Dong, Jerry Qu, Gideon Schmidt

10 points:
Alicia Chen, Philip Sun, William Wang, Alex Wei

High scoring newcomer award:
Sean Dory

"Whole is more than the sum of the parts" team award:
Daniel, Nabihah, Swetha, Thomas, Yang

## Monday, December 13, 2010

### In memoriam: Joel Brainard

Our thoughts and hearts go out to Joel's daughter, Katherine Brainard, a founding member of our math circle, and to Joel's wife, JC Glendinning, an important driving force behind the beginnings of our math circle in 2001, and to their entire family.

Joel Pennington Brainard, 71, died peacefully on December 11, 2010, at NY Presbyterian Hospital in New York, NY. He is survived by his wife Jane Carol Glendinning; their children, Junior, 29, Katherine, 25, and Scott, 23; and his brothers Charles and William. A graduate of Oberlin College, Joel served in the Peace Corp in the Ivory Coast and later taught mathematics at Talladega College in Alabama. With advanced degrees from MIT and Cornell University, Joel was an engineer on energy conservation projects at Brookhaven National Laboratories on Long Island and was a consultant on public utility issues for the Vermont Low-Income Advocacy Council through Vermont Legal Aid in Burlington. His many colleagues remember him fondly for his more than 22 years of service as an economist and manager in the Office of Research at the NYS Public Service Commission. Joel was an extraordinary man whose thought, wit and kindness touched the lives of all who knew him. Family and friends remember his joy for life with deep affection. His energy and enthusiasm permeated his life, particularly his ingenious solutions of problems, both large and small. From reducing distortions in the pricing of the electrical grid to his unique approaches to car repairs, home construction, and even ski boot insulation and golf cart-to-ATV modification, Joel provided an inspiring example and will be greatly missed. In lieu of flowers, the family requests that gifts be made in Joel's memory to the Robert C. Parker School, which Joel helped found 20 years ago, or the Scleroderma Foundation, an institution which works to find treatments for an autoimmune condition that Joel fought creatively for many years. A Memorial Service will be held on Saturday, December 18th at 11:00 am at the Robert C. Parker School, 4254 West Sand Lake Rd, Wynantskill, NY.

## Sunday, December 12, 2010

### The biggest Evil Abundant Number submitted at today's Middle School Math Meet?

The teams at our middle school math meet today submitted the following numbers as candidates for the "biggest evil abundant number you can find."

24
720
111100
and

$12^{10000000^{1000000000^{100000000^{10000000^{100000000000^{1000000000000^{10000000}}}}}}}$

And the winner is.....well, that's not so clear. We'll discuss it below. We'll also tell you what evil numbers and abundant numbers are.

But, first, a few important words of thanks!

We had a GREAT Middle School Math Meet today! Thanks very much to Felix Sun, Qun Lu, and the Principal of the CCC Chinese School Jianzhong Tang for making arrangements to host our December Middle School Math Meet at Shaker Junior High this afternoon. Thanks as well to Felix's mother, Le Xu, for organizing refreshments!

Thanks to UAlbany Professor Rita Biswas, Hebrew Academy math teacher Alexandra Schmidt, and Doyle Middle School math teacher Nancy Smith for helping to run the Math Meet. Thanks as well to our outstanding high school student coaches: Felix Sun (Shenendahoah High School), Zubin Mukerjee (Guilderland HS), Cecilia Holodak and Flora Mao (Niskayuna HS), Simran Rastogi and Gili Rusak (Shaker).

Thanks to all the students who came and worked enthusiastically on the problems.

Thanks to George Reuter of mathmeets.com, who did a great job of writing more great contest problems for the December math meet. We can't discuss those questions yet, since other teams may still be taking that contest.

Okay, so back to this evil and abundant question, which we CAN discuss, since I just created it as a little supplementary challenge to fill in the bits and pieces of waiting time during the meet. It turned out to be way more interesting than I had realized!

We began the Math Meet by discussing the "number of the day: 12." (Why, because it is December 12, or 12/12, of course!)

Like all numbers, 12 has many interesting properties. We focused on two of those properties today, which are highlighted in the Tagxedo-produced graphic above.

Twelve is an "evil number", which means that it has an even number of ones in its binary expansion, i.e., 1100base 2 = 1*8 + 1*4 + 0*2 + 0*1 = 12.

Twelve is also an "abundant number," because 12 is less than the sum of its proper factors, i.e., 12 < 14 = 1+2+3+4+6. In fact, it is the smallest abundant number, and therefore, of course, it is also the smallest evil abundant number, as well.

Is there a largest evil abundant number?

Some students noted that doubling an evil number always gives you another evil number! (Why?) What if you tripled an evil number? Or multiplied your evil number by other integers? What if you add two evil numbers? What if you raise an evil number to a power? Do you always get another evil number?

(By the way, you could ask the same questions about "odious numbers," which are the opposite of evil numbers--they have an odd number of ones in their binary expansions.)

Other students noted that doubling an abundant number always gives you another abundant number! (Why?) What if you tripled an abundant number? Or multiplied your abundant number by other integers?

Again, you could ask the same questions about perfect numbers, or deficient numbers.

More interesting questions: can a power of two ever be an evil number? Why or why not? Can a power or two ever be an abundant number? What about powers of three?

All great questions to think about!

Now back to judging the entries submitted in the contest.

24 is clearly evil (binary representation is 11000) and abundant (its proper factors are 1,2,3,4,6,8,12, which sum to more than 24.)

720 is also evil (binary representation is 1011010000) and abundant (its proper factors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360).

111100 turns out to be trickier. It is not clear whether it was intended to be interpreted as a binary number, in which case it would obviously be evil. If it was intended to be interpreted as a base-10 number, then it's actually not evil, because the binary representation of 111100base 10 is 11011000111111100. So the only way this number is a candidate is if we interpret 111100 as intended to be binary, in which case it is only equivalent to decimal 60. Now, 60 is clearly abundant, but it is less than 720.

Now, for that humongous number with all the concatenated exponents. I can tell you it is abundant, for sure, but I have no idea if it is evil or not. I am not convinced that the team that submitted it is sure whether it is evil or not, but if they can come up with a convincing proof that it is evil, I am willing to listen. Thinking systematically about some of the questions I raised above may help you sort out what is going on with your--very interesting--number. You may also want to consult the following book, which introduced the concepts of evil and odious numbers: Winning Ways for Your Mathematical Plays Volume 3, by Elwyn R. Berlekamp, John Horton Conway, and Richard K. Guy.

In the meantime, until such time as the orange team can demonstrate their candidate number is evil, 720 is the winner among the numbers submitted today.

Congratulations to the Green Team: Sean, Jason, Gideon, Frank, and Aaron. Your entry of 720 is the largest confirmed evil abundant number of those submitted today.

And, everyone, keep on thinking about this problem! Can you come up with a formula for an arbitrarily large evil abundant number that you can PROVE will work?

## Tuesday, November 23, 2010

### Princeton University Math Contest (PUMaC)

Teams from around the country and as far away as Beijing, China competed at the fifth annual Princeton University Math Contest (PUMaC) this past weekend.

Congratulations to the eight Albany Area Math Circle students who did an outstanding job of representing our math circle at PUMaC! Pictured below are team members Jay White (heeg), Felix Sun (Shenendahoah), Gili Rusak (Shaker), Ashley Cho (Emma Willard), Gurtej Kanwar (Bethlehem), Matthew Babbitt (heeg), Zubin Mukerjee (Guilderland) and Preston Law (heeg).

The team tied for fourth place in the highly competitive A-division team round, tying with top teams from such distinguished places as Phillips Exeter Academy, New York City, and Beijing.

PUMaC team captain Matthew Babbitt and senior veteran Ashley Cho both brought home individual A-division medals as well on the geometry round, with Matthew tying for 4th place and Ashley tying for 7th place.

There was much friendly camaraderie in the cars travelling to and from the contest, as well as games during the breaks in the competition day. Team member Zubin Mukerjee won the chess mini-event during the lunch break.

A special highlight this year was the opportunity for team members to visit with Andrew Ardito and Dave Bieber, much beloved alumni of our math circle who are now Princeton undergraduates helping to run the contest.

## Saturday, November 20, 2010

### Wise and heartfelt words from a veteran

Kate Rudolph, one of last year's top ten winners at the 2009 Math Prize for Girls, and now an MIT freshman, came back to speak with this year's participants at the 2010 Math Prize for Girls award ceremony, held in the historic Cooper Union Great Hall.

The gold-fringed velvet covered podium in the Great Hall is an imposing and intimidating place for a young woman to speak, even one as accomplished as Kate. (MIT describes her as a "world class mathlete.")

In its 150 years of history, "rebels and reformers, poets and presidents" have spoken at that podium. Many great orators--Abraham Lincoln, Mark Twain, Teddy Roosevelt, Barack Obama, the leaders of the antislavery,civil rights, and women's suffrage movements have all made impassioned speeches in that hallowed hall. Not exactly easy acts to follow!

Kate stepped up to the challenge with poise and passion, defying the stereotypes and demonstrating her gifts with words as well as numbers. Her enthusiastic words of encouragement and advice resonated throughout the Great Hall, giving new perspectives to the hundreds of girls and parents anxiously awaiting the results of the competition.

Here is an excerpt from the inspiring, passionate, and thoughtful speech Kate had clearly put her heart into composing to share with the 2010 contestants.

[A]ll of these competitions, including the one you took today, have something in common: you’re solving “solved” problems. Someone else has already thought about the problem and knows the solution before you even get to see it. Imagine what it must feel like to be the first person, in the world, to solve a math problem.

Well, that’s what the future holds for all of you. The point of the Math Prize for Girls is to get more girls interested in math, but what for? Moving forward, the point is NOT to get you winning more math contests. It’s not even to get you writing good math contests (which is considerably harder.) The point of bringing you here and exposing you to these problems is to get you interested enough in math that eventually you want to do NEW MATH.

But how do you get there from here? You’re on the right track: seeking out challenging problems and pushing yourself to do well. What’s next? I have three main pieces of advice.

The first: work hard. Nothing worth achieving is easy to achieve. Math is not easy, and it gets harder, but you can solve any problem you come across with enough effort.

Next: do what you love. At some point during your life, there will come a time when you break down, or burn out. If you spend your time doing what you love, you will have the foundation to build yourself back up again.

Finally: work with other people. This seems sort of opposite to the concept of competition math, where you’re primarily working individually, but I can tell you that there are math problems that are so hard, if you didn’t combine forces with other people you wouldn’t get anywhere.

Look around. You may know some of the other contestants here: you may have met them in summer math programs or they may be from the same area as you. But I challenge you not to just let this be a reunion of people you already know, but a chance to meet more math girls and expand your network. When I say “work with other people,” THESE are some of the best “other people” you could find!

Finally, I think I need to acknowledge that yes, we girls are a minority in the field of mathematics. However, this is not a curse but a blessing, and I challenge you to use it! As a girl, you will have more opportunities (heck, you’re taking advantage of one of them right now!) If people set lower expectations for you because you are a girl, that is an opportunity to blow them away.

Good luck on this competition: I know you’re all anxiously waiting for the results. But more importantly, good luck as you pursue mathematics beyond this competition, beyond all competitions, and start doing some NEW MATH of your own."

## Friday, November 19, 2010

### More fun and games at Math Prize for Girls

'Twas the night before the Math Prize for Girls, and some girls sat on the floor to build surprisingly sturdy structures from marshallows and spaghetti, while others played board games. There's a lot to be said for cultivating a relaxed playful mind--playfulness is an important part of problem-solving.

The challenging math problems that awaited them the next morning are available here.

## Tuesday, November 16, 2010

### Math Prize for Girls contestants community: who are they and where are they from?

Thanks again to Advantage Testing Foundation for its vision and generous lead sponsorship of the Math Prize for Girls in 2009 and again this year. Thanks as well to the additional sponsors who have now joined Advantage Testing Foundation in supporting the event: Akamai Foundation, Canada/USA MathCamp, and Wolfram Mathematica.

This past weekend's Math Prize for Girls event provided even more opportunities for girls from all over the United States and Canada to forge new friendships and renew old ones based on their shared enthusiasm for mathematical challenges. This year's event added a lunch for contestants on Saturday as well as a "Games Night" event Friday evening for distant students whose travel plans had them staying overnight in the Big Apple.

There's a wonderful camaraderie of kindred spirits at all the mathematical events I've ever attended, but the atmosphere at the Math Prize for Girls stands out even more for all the sparkling smiles and laughter and friendly hugs. Students did not even seem to mind the long lunch lines--taking that as yet another opportunity to make new friends they might not otherwise have met. The purple hats and t-shirts were apparently major hits.

Veteran students returning for a second year welcomed the opportunities to reconnect with friends they met last year, as well as to make new friends with first-timers. Students, volunteers, and fans who knew one another from ARML or other contests like HMMT or summer programs such as Canada/USA MathCamp and HCSSiM reunited and connected up their intersecting networks of friendships.

For students from remote outposts who had never participated in a math event outside their own schools, this event was a particularly special opportunity to make new friendships with girls who shared their passion for mathematical challenges. Some girls reported that they had so much fun talking to students from all over that they have been inspired to form local math circles to recruit more kindred spirits to work together to keep that enthusiastic mathematical community experience going throughout the year in their hometowns.

So, how about a little friendly rivalry?

Which states and provinces had the most girls participating this year?

Top States and Provinces
Based on Raw Numbers of Participants

1) California and New York (tie 28 girls each)
2) New Jersey 22 girls
3) Massachusetts 18 girls
4) Connecticut 16 girls
5) Illinois 9 girls
6) Pennsylvania and Virginia (tie 8 girls each)
7) Maryland 7 girls
8) Georgia, North Carolina, and New Hampshire (tie 6 girls each)
9) Michigan, Texas, and Washington (tie 5 girls each)
10)Indiana and Ontario (tie 4 girls each)
11)Florida and South Carolina (tie 3 girls each)
12)Iowa, Missouri, Ohio and Wisconsin (tie with 2 girls each)

Other states and provinces represented at 2010 Math Prize for Girls: Alberta, Arizona, British Columbia, Colorado, Kentucky, Manitoba, New Mexico, Utah (1 each)

Of course, reasonable people may object that the rankings above do not take into account population or distance from New York City. It's hardly fair to compare New York State to Manitoba!

So I generated some alternative rankings below by adjusting for population and/or distance, using Wolfram Alpha's handy data retrieval features to get statistics on the distance of each state or province's capital from New York City as well as its population.

Top States and Provinces at Math Prize for Girls

(Ranking based on raw numbers divided by state or province population)

1) New Hampshire
2) Connecticut
3) Massachusetts
4) New Jersey
5) New York
6) Maryland
7) Virginia
8) Manitoba
9) California
10) Washington
11) Illinois
12) Iowa

Top States and Provinces at Math Prize for Girls
Adjusted for Distance from New York City

(Ranking based on raw numbers multiplied by number of miles from state or province capital to NYC)

1) California
2) Washington
3) Texas
4) Illinois
5) Georgia
6) New York
7) Massachusetts
8) Michigan
9) Florida
10)Indiana
11)North Carolina
12)British Columbia

Top States and Provinces at Math Prize for Girls
Adjusted for Population and Distance from New York City

(Ranking based on raw numbers multiplied by miles from capital to New York City divided by state or province population)

1) California
2) Washington
3) Manitoba
4) New Hampshire
5) New Mexico
6) Utah
7) Iowa
8) Alberta
9) British Columbia
10) Illinois
11) Massachusetts
12) Georgia

Of course, reasonable people can differ about the appropriate methodology. Perhaps it is better to multiply by the square root or logarithm of the distance in miles to take into account the fact that travel times are not linear in distance. Perhaps states with boarding schools that recruit from national or international pools should have an adjustment to the population divisor to take that into account.

Can you come up with a methodological approach that puts your state or province on top? Feel free to post it in the comments below.

Put your state or province on the mathematical map next year!

And a shout-out and note of friendly encouragement to students in states and provinces that did not participate at all this year: if your state or province is small and/or far-away, it would only take one or two participants to make it rise to the top of the adjusted rankings above!

What's the first step?

If your high school does not offer the Mathematical Association of America's American Mathematics Competitions AMC10/12 contests offered each February, this is an excellent time to approach the head of your school's math department to ask about taking it. The AMC10/12 can lead to many opportunities--not just the Math Prize for Girls, but also the American Invitational Math Exam, the USA Math Olympiad, the Math Olympiad Summer Program, and even the International Math Olympiad as well as the China Girls Math Olympiad.

The per student cost for the AMC10/12 is very reasonable if you can persuade a number of your friends to join you in taking it. If your school takes advantage of the early registration discount and signs up before December 18, it works out to less than $6 per student if ten students take it at your school, about$2 per student if 100 students take it, and the cost goes asymptotically down to \$1.60 per student if an infinite number of students take it at your school!

Best of all, you and your fellow students can have a lot of fun and learn a lot by preparing together. The best way to learn is to help others prepare as well. Create a mathematical community at YOUR school! Here is a resource website I've put together to help students prepare.

Photo credits: Joy Mingalingading (photo at top); Dr. Madhu Boppana (other photos)

## Monday, November 15, 2010

### Congratulations Ashley!

Albany Area Math Circle veteran So Yeun (Ashley) Cho, a senior at Emma Willard School, brought home yet another beautiful sparkling crystal trophy this year for a performance even more outstanding than last year in an even more stellar field of over two hundred extremely mathematically talented young women from all over the country and Canada at the Math Prize for Girls held at the Courant Institute of Mathematics on Saturday.

This year, Ashley was tied for 18th place, only one question away from top 10 tiebreak standing! Last year, she was tied for 25th place, two questions away from the top 10.

Thanks to her Honorable Mention score at the Math Prize for Girls this year, Ashley will be invited to write the Math Prize Olympiad in December, a four hour olympiad-style contest with proof-based problems designed to be comparable to those on the China Girls Math Olympiad.

In both 2009 and 2010, Ashley's score at the Math Prize for Girls was the highest for anyone from New York State! The complete list of high scorers is available here.

This year's field of over 200 competitors was even stronger than last year's field, because the success of last year's event caused much greater interest and many more girls from all over the country applied to participate. The event has attracted additional major new sponsorship this year, Akamai Foundation, which also sponsors the Math Olympiad Summer Program and the US team to the China Girls Math Olympiad. Thanks to additional sponsorship, more exceptionally strong students from distant locations were able to participate this year, because all 2010 USAMO qualifiers received travel subsidies. In addition, the Math Prize for Girls accepted applications from Canadian students for the first time this year. Thus, Ashley's stronger performance this year is all the more impressive, since the field is even stronger.

Ashley has been an outstanding leader and inspirational model for her fellow students at Emma Willard and for all students in our math circle. She has also been a leader among all students in New York, with many other recognitions including qualifying for the USA Math Olympiad in 2008, team high scorer honors on our A team at the state math tournament, NYSML, last April, selection for the Upstate NY ARML A team where she contributed to the epic first-place super relay in 2009, and selection to participate in Princeton's Summer Workshop in Mathematics last summer.

Ashley has mentored and coached her fellow students on Emma Willard's Harvard-MIT Math Tournament (HMMT). Next weekend she will join seven of our math circle's strongest and most experienced students to represent Albany Area Math Circle at PUMaC (Princeton University Math Contest), an extremely challenging math contest.

Our first-timers at Math Prize for Girls this year were Sherry He and Wan Wan Fei, also from Emma Willard, Cecilia Holodak and Elizabeth Parizh from Niskayuna High School, and Gili Rusak from Shaker Junior High School. The photo at the top of this post shows Wan Wan, Albany Area Math Circle founding member Alison Miller, and Sherry celebrating along with Ashley.

Congratulations to all the participants in Math Prize for Girls this year, from Albany Area Math Circle, from all over New York State, from all over the country and Canada, too!

## Thursday, November 11, 2010

### Harvard-MIT Online Tournament at Emma Willard 11/7/10

Congratulations to the brave group of Albany Area Math Circle students who spent five hours last Sunday afternoon working on an "Extreme Math" challenge--the Harvard-MIT Online November Tournament. Though most of the students were new to this kind of intense mathematical experience, they rose to the challenge with great enthusiasm and camaraderie. We had three composite teams bringing together students from Doane Stuart, Emma Willard, Guilderland High School, and Niskayuna High School.

To honor our region's grand traditions of mathematical and scientific contributions, our three teams were named after three historic scientists from our area: Charles Steinmetz, Katherine Blodgett, and Frank Benford. Charles Steinmetz, the mathematical wizard of Schenectady was a mathematician and electrical engineer who started the GE Research Laboratories, served as President of the Schenectady School Board, and taught at Union College. It was said that he figured out a way to turn imaginary numbers into electricity. Katherine Blodgett was a Schenectady native who was the first woman ever to get a PhD in physics from the University of Cambridge, and who later worked at GE where she invented a molecular process to create the world's first 100% invisible glass. Frank Benford was an electrical engineer and physicist now best known for his discovery of Benford's law, an unexpected but beautifully surprising logarithmic statistical pattern that can be used to detect financial frauds and other fabrications. (The story behind his discovery of the pattern is delightfully serendipitous: while working at the GE Research Laboratories in Schenectady, he noticed that the pages of the logarithmic tables corresponding to numbers with a leading digit of 1 or 2 were much dirtier than the pages for numbers beginning with 8 or 9, which suggested to him that numbers beginning with 1 or 2 might occur much more frequently in real world data than numbers beginning with 8 or 9.)

The students demonstrated remarkable concentration and perseverence throughout the long hours of the multi-round mathematical contest--I believe that Steinmetz, Blodgett, and Benford would be very proud of the strong and persistent efforts of the students on their namesake teams.

Students on Team Steinmetz (Blue) were: Aniket Tolpadi, Elizabeth Parizh, George Gelashvili, Peggy Hsu, Sherry He, and Yvonne Yen. Students on Team Blodgett (Red) were: Candice Chiu, Jamie Park, Justina Liu, Luxi Peng, and Sunny Yan. Students on Team Benford (Green) were: Chelsea Wu, Claire Feng, Eric Dammerman, Isaac Malsky, and Shuang Jin.

Team High Scorers:

General Round:
Elizabeth Parizh
(Steinmetz Blue)
Candice Chiu (Blodgett Red)
Chelsea Wu (Benford Green)

Theme Round:
Sherry He (Steinmetz Blue)
Justina Liu (Blodgett Red)
Isaac Malsky (Benford Green)

High scorers of the meet
(based on combined total on general and theme rounds):
Sherry He (first place)
Chelsea Wu (second place)
Elizabeth Parizh (third place)
Honorable Mentions: Aniket Tolpadi, Justina Liu, Luxi Peng

Team Round Local Winner: Blodgett (Red)
GUTS Round Local Winner: Blodgett (Red)

Whole is More than the Sum of the Parts Award:
Benford (Green)

Spirit of the Meet Award: Peggy Hsu
for her awesome enthusiasm and encouragement to our many first-time participants

Many thanks to Emma Willard School for agreeing to host this very enjoyable opportunity for our newer students to really immerse themselves in a lengthy and challenging mathematical experience!

Special thanks to Emma Willard math department head Sunshine Greene for working with us to make all the advance arrangements, as well as to Emma Willard teachers Judy Price, Carmel Schettino, Meredith Legg, and Angela Richard for all their help running the event.

Thanks as well to Emma Willard Practicum Coordinator Anne Mossop for all she has done to support Emma Willard students in making connections with our math circle on Friday evenings! Emma Willard students been a tremendous and growing addition to our math circle in recent years--and we are delighted to have their talents and enthusiasm among us.

Special recognition is due to Emma Willard senior veteran Ashley Cho for all her leadership and encouragement of Emma Willard students, as well as her help in getting all the contestants to the right place at the outset of our contest.

Finally, a shout-out to Beth Schaffer, former Captain of Albany Area Math Circle, now an MIT senior. Beth launched new initiatives--the November tournament and the on-line version of the November and February tournaments--during her tenure as HMMT tournament co-director for the past two years. Both these innovations reflect her concern for outreach and inclusion.

## Thursday, November 4, 2010

### AMC8 Tuesday Nov 16

The AMC8 is coming up on Tuesday November 16. 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 the Capital District.

Albany Area Math Circle is grateful to Hebrew Academy of the Capital District for its hospitality!

## Sunday, October 31, 2010

### Try again! Fail again! Fail better!

Students: Were you feeling discouraged by the difficulty of the problems you tried at our math circle on Friday evening?

If you found yourself struggling and flailing about and feeling lost, that means you are in the right place!

The problems we work on at math circle are supposed to be a LOT harder than the problems you get in regular school classes. Struggling, flailing about, feeling lost a lot--that is what mathematics is all about.

Mathematician Lillian Pierce has great advice on learning mathematics in an interview in the latest issue of Girls Angle Bulletin:

Enjoy math!

Especially, enjoy challenges!

Math is just like any other skill: practice counts.

Also, have courage and confidence in your abilities. Don't shy away from failing to solve something immediately: try out your ideas, make mistakes, learn from your mistakes.

The struggle itself is one of the most important parts of your practice.

Try again!

Fail again!

Fail better!

When you have been beating your head against a problem for a while, and getting nowhere, leave it and move on to other problems, struggle with those for a while, then come back with a new perspective. Sometimes just finding a partner so you can share what you've been trying can help a lot. Even if your partner is just as lost as you are, the simple act of explaining aloud what you've been trying may help to clarify the problem in your own mind. And perhaps you have a great approach but finding a small blind spot your partner can help you spot will make all the difference.

Remember: practice counts! And practice means making mistakes! It is OKAY to make mistakes, lots and lots of mistakes at our math circle meetings. If you are NOT making mistakes at our meetings, you are doing the wrong problems!

Don't be embarrassed about making mistakes at our meetings! All of us make mistakes. Celebrate your mistakes! Learning from your mistakes, your half-baked ideas, your false starts--that is the essence of problem solving.

Our math circle is a place where it's really, really okay to make mistakes! Don't be afraid. Enjoy that space and freedom to make mistakes! It's a treasure! Share your mistakes and what you've learned from them.

## Friday, October 29, 2010

### Signup for Harvard-MIT Math Tournament LOCAL

Thanks to Emma Willard School for their gracious hospitality in accommodating our math circle's local administration of this November contest to our high school student members.

## Monday, October 18, 2010

### "maverick mathematicians" and the power of grade school geometry

Earlier this week, the New York Times published this obituary for Benoît Mandelbrot, excerpted below:

Benoît B. Mandelbrot, a maverick mathematician who developed the field of fractal geometry and applied it to physics, biology, finance and many other fields, died on Thursday in Cambridge, Mass. He was 85. ....

Dr. Mandelbrot coined the term “fractal” to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature.

“Applied mathematics had been concentrating for a century on phenomena which were smooth, but many things were not like that: the more you blew them up with a microscope the more complexity you found,” said David Mumford, a professor of mathematics at Brown University. “He was one of the primary people who realized these were legitimate objects of study.”

In a seminal book, “The Fractal Geometry of Nature,” published in 1982, Dr. Mandelbrot defended mathematical objects that he said others had dismissed as “monstrous” and “pathological.” Using fractal geometry, he argued, the complex outlines of clouds and coastlines, once considered unmeasurable, could now “be approached in rigorous and vigorous quantitative fashion.”

For most of his career, Dr. Mandelbrot had a reputation as an outsider to the mathematical establishment. From his perch as a researcher for I.B.M. in New York, where he worked for decades before accepting a position at Yale University, he noticed patterns that other researchers may have overlooked in their own data, then often swooped in to collaborate.

...

Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain? The answer, he was surprised to discover, depends on how closely one looks. On a map an island may appear smooth, but zooming in will reveal jagged edges that add up to a longer coast. Zooming in further will reveal even more coastline.

“Here is a question, a staple of grade-school geometry that, if you think about it, is impossible,” Dr. Mandelbrot told The New York Times earlier this year in an interview. “The length of the coastline, in a sense, is infinite.”

At last week's middle school math meet, we talked about a special kind of fractals, Pythagoras trees. The particular type of tree we investigated was made entirely out of squares and isosceles right triangles.

We started with the assumption that the biggest square in each fractal was 64, and tried to figure out how the area of the fractal grows as we add layers. One way to solve this problem is to use the Pythagorean theorem over and over and over again. That is the rather laborious approach that teachers came up with this summer at the Bard summer workshop for middle school math teachers.

Max Thomas of Hackett Middle School, however, came up with a very clever alternative approach. He noted that you can figure out that the isosceles right triangle in B1 must have area 16, because you can mentally "fold" it down into the big square below it, and easily confirm that it has area equal to one fourth of the big square. By similar reasoning, you can fold the same triangle in B1 over into the smaller squares in B1 and see that each of those two smaller squares must have an area of 32.

So, the total area of the fractal in B1 must be 64 + 16 + 32 +32 = 144. Can you extend Max's approach to figure out the areas of the succeeding fractals, and the general pattern as you add each layer of the fractals?

You can also investigate other kinds of Pythagorean trees by downloading free interactive applications from Wolfram MathWorld here.

Wolfram alpha also offers many other nice interactive fractal demonstrations here.

As Professor Mandelbrot and Max Thomas have both demonstrated, you don't always need to use high-powered mathematics to get very deep insights into fractal mathematics. Sometimes you don't even need the Pythagorean theorem, though playing with more complicated Pythagorean trees will certainly motivate more use and understanding of that idea. And if you persevere, the study of fractals can take you into more and more complex mathematics.

But first things first. Start with simple grade-school geometry and the Pythagorean theorem. See how far they can take you.

See what you can discover by investigating fractals playfully, starting with simple cases and working you way into the general patterns.

## Thursday, October 14, 2010

### Math Marketing Done Wrong!

Dear parents of students who attended our first middle school math meeet,

Thank you again for bringing your children to our first middle school math meet! They were a thoroughly delightful group to work with, and they helped our year of math meets get off to a great start!

It has come to our attention that Aileen Leventon, who is apparently a marketing person for a private tutoring business called "Math Done Right," was soliciting parents waiting to pick up their children after the meet, with information promoting that company.

From our perspective, it was "Math Marketing Done Wrong!"

Ms. Leventon did not have our permission or endorsement to market her business to you, nor did she have permission from Hebrew Academy to market her business to you.

Please know that Albany Area Math Circle adult advisors are VOLUNTEERS! We are volunteering our time to create a vibrant mathematical community of kindred spirits who love problem solving together.

We are NOT volunteering our time to provide a marketing platform for a private business about which we know absolutely nothing.

If you encounter anyone attempting to solicit you for a private business at an Albany Area Math Circle event, please bring it to the attention of an Albany Area Math Circle advisor immediately. You can recognize us by our "Tough Traveler" ID badges (see below.)

Mary O'Keeffe

## Tuesday, October 12, 2010

### Thanks to Hebrew Academy of the Capital District ...

... for hosting students from schools all over the Capital District at our October 2010 Middle School Math Meet!

Names and scores of high-scoring students and the value-added whole-is-more-than-the-sum-of-the-parts team award will be posted in this space later this week after the administration window ends.

Thanks to George Reuter and Mike Curry of MathMeets.com for writing a great set of problems for our students! Thanks to student coaches Matthew Babbitt and Zubin Mukerjee as well as Albany Area Math Circle advisors Bill Babbitt and Rita Biswas for helping make the meet great. A special thanks to Albany Are Math Circle Advisor and Hebrew Academy math teacher Alexandra Schmidt for all the arrangements she made for us.

Would YOUR school like to host our next monthly math meet, scheduled for Sunday November 7? If so, please get in touch with us by sending an email to mathcircle@gmail.com.

In addition to working on those Math Meet problems, we also talked about fractals and other fun mathematical topics today.

In honor of the recent 10/10/10 day, we talked about powers of ten, powers of two, and 42.

For example, we talked about how 101010 in binary is 32+8+2=42 in decimal, and of course how 42 is a delightfully special number.

(In addition to being the Answer to the Ultimate Question of Life, the Universe and Everything, we noted that 10!seconds = 42 days (exactly!) because

[10x9x8x7x6x5x4x3x2x1 seconds]÷[(24 hours/day)x(60 minutes/hour)x(60 seconds/minute)]=42 days.

We also introduced the concept of "bimal" notation, which is the binary analog of decimal notation.

In bimal, 1/2 = 0.1, 1/4 = 0.01, 1/8 = 0.001, 1/16 = 0.00001, and so on.

This allows us to nicely express the following question:

What is 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + .... (and so on forever)?

In bimal, the answer is 0.111111111111111..... (and so on forever).

But, just as in decimal, 0.999999999...... = 1, in bimal 0.111111111.... = 1.

And we talked about a special kind of fractals, Pythagoras trees. In honor of Dr. Benoît Mandelbrot, who died a few days after our middle school math meet, I have moved our discussion of that topic to a separate post, Maverick Mathematicians and the Power of Grade School Geometry.

## Monday, September 27, 2010

### Middle School Math Meet Tuesday October 12

We again encourage our high school students to coach MATHCOUNTS teams and/or to organize and lead small middle school math circles for students in their areas.

Albany Area Math Circle is also offering something new for middle school students this year.

We will have a special monthly "Middle School Math Meet" event once a month.

Our first "Middle School Math Meet" event will be hosted by the Hebrew Academy of the Capital District on Tuesday October 12.

The meet will officially start at 4:15, but we will offer warmup activities for students who can arrive at 3:45. The meet will end by 5:30. Refreshments will be served. Students should bring pencils with good erasers. Please leave your calculators at home!

We will rotate days and locations for this monthly event in future months, including Sunday afternoons as well as weekday afternoons, so don't be discouraged if Tuesday afternoons are not good for you.

We also plan to rotate locations for this event, so please let us know if you would like to arrange for your school to host a future math meet. A special thanks to Hebrew Academy of the Capital District for "breaking the ice" and hosting the first one!

Parents of students who would like to participate in the October 12 math meet should use the form at this link to register their student.

## Saturday, September 25, 2010

### Princeton University Math Contest (PUMaC)

Albany Area Math Circle plans to participate in PUMaC again this year. This is a very challenging and exciting experience run by the undergraduate math students at Princeton University each year since 2006.

(It's exciting that two of our alumni are now undergraduate students at Princeton. Both Andrew Ardito and Dave Bieber participated in PUMaC throughout their high school years, and I'm sure they and all the students involved are eager to draw on their experiences to contribute their ideas to make this year's contest even better.)

Albany Area Math Circle teams have done very well in the past, and we plan to send two teams this year.

Here are the key dates that Albany Area Math Circle students need to keep in mind for PUMaC:

As soon as possible, email me at mathcircle@gmail.com to let me know of your interest, even if you are not yet entirely sure of your schedule and ability to commit. We'll be happy to answer questions and to put your parents in touch with other parents whose students are considering going so they can discuss possible carpools and hotel-room sharing arrangements, etc. among themselves.

Sunday October 10: Deadline to inform me of a firm commitment to participate.

Saturday November 13 through November 19: Power Round of PUMaC All 8-person teams will work collaboratively during this time to solve and write up solutions to a multipart Power Round problem. Math circle students will do this collaboration mostly via Internet, and it's important that students who sign up for PUMaC make sure they will have time to work on the problems during this period and contribute to the solution and writeup.

Saturday November 20: All day on-site contest at Princeton University. The Power Round will be turned in at the beginning of the day. The remaining rounds (Individual Subject Tests and collaborative Team Round Test) will take place this day, along with a number of just-for-fun "Mini-Events" that don't count toward the final team ranking (Math Bowl, Rubiks Cube, Useless Math Olympiad, Chess, etc.)

For more information, you may want to talk to students who have gone in the past about their experiences. Veterans from last year who have already notified me of their enthusiastic plans to return this year are: Matthew Babbitt, Gurtej Kanwar, Zubin Mukerjhee, and Jay White. Ashley Cho is also definitely going. Once we have the team composition settled, we will hold a preliminary organizational meeting to help our first-time participating students prepare to make the most of this exciting opportunity.

I have created a special google groups email discussion list for Albany Area Math Circle students who are possibly interested in going to PUMaC this year. Even if you are not sure if you can go yet, you are welcome to join the group to find out more about preparing for PUMaC. Email me as soon as possible if you are interested in PUMaC for this year and I'll add you to the list.

UPDATE: See comments below for questions and answers from math circle students about how PUMaC is going to work.

## Wednesday, August 4, 2010

### Andrew Ardito: Soul of the Albany Area Math Circle

Andrew Ardito, our math circle's captain for the past three years, has racked up an extraordinary string of accomplishments during his time with our math circle. His infectious passion for problem solving challenges has inspired many others in our math circle community, both younger and older, and we will all miss him enormously as he heads off to college this fall.

Andrew is a homeschooled student from Coxsackie whose accomplishments go well beyond mathematics. By the end of his junior year of high school, he had already reached National AP Scholar status by getting high scores on nine AP exams, and he took five additional AP exams in his senior year. Those 14 AP exams covered a wide variety of disciplines including literature, art history, US and world history, economics and government, as well as mathematics and all the natural sciences. He also managed to find time to study four years of Russian during high school. He went into special depth in mathematics and physics, taking several advanced physics classes at SUNY Albany as well as even more advanced math classes at SUNY Albany and RPI, including number theory, topology, analysis, and a graduate class in combinatorics. His many honors also include a National Merit Scholarship and a Robert C. Byrd Scholarship. In addition, Andrew has won numerous awards for chess, as well as coaching a chess club for younger students.

Andrew started out as one of the youngest members of our first middle school math circle, back in 2003. He eagerly moved on to greater and greater challenges, taking over as captain of our high school math circle in his sophomore year of high school. He also student-coached MATHCOUNTS teams throughout his high school years. In his junior year, he helped spur a reinvorigation of our middle school math circles, and he leaves a remarkable legacy of inspiring leadership behind, as he has worked with so many students who have already demonstrated great promise as mentors themselves.

He is an outstanding example of the guiding principle of our math circle: the best way to learn is to share what you think you already know with someone else, because explaining it to them deepens your own understanding.

Andrew has accumulated an extraordinary number of accomplishments during his time as a member of our math circle, both in mathematics and in other areas.

A partial list of his many honors in mathematics include:

☞ National MATHCOUNTS contestant all three years as a member of the New York State team in middle school. As a 7th and 8th grader, he won the New York State championship and made the National Countdown Round in both years, placing sixth on writtens in 2005 and and 2nd place in the country at MATHCOUNTS Nationals Countdown in 2006. He also provided team leadership to two Chapter-winning teams, one of which came in 3rd place in the state.

☞ As a high school student, Andrew coached many successful MATHCOUNTS teams, including several teams that won Chapter championships and one team that won a state championship. He also coached many students who won high individual honors at Chapter and State MATHCOUNTS contests, including quite a number of Chapter and State CountDown participants and National MATHCOUNTS contestants. His mentorship and enthusiastic example has also inspired many younger students to embrace some exciting and challenging high school contests while still in middle school, including AMC10/12, HMMT, and NYSML/ARML.

The photo above shows Gili Rusak, a 7th grader at Shaker Junior High, and Aniket Tolpadi, an 8th grader at Iroquois Middle School, working with Andrew in a middle school math circle meeting. Both Gili and Aniket qualified for the American Invitational Math Exam (AIME) this year. Gili was the only 7th grader in the state to qualify for this exam, while Aniket was one of only four 8th graders in the state to do so.

☞ Andrew has earned many outstanding honors in high school math contests: he began taking the AMC high school contests in sixth grade and has won many honors in that series: he qualified for the AIME seven times, for the USA Math Olympiad five times. In 2007, he qualified for the Math Olympiad Summer Program; in 2008, he was the top scorer in the state on the AMC12B, and he also won an AMC12 gold medal for his consecutive four years as math circle's high scorer on the 12A and/or 12B.

Andrew has consistently been among Albany Area Math Circle's top three scorers on the AMC12 in every year since he began taking that exam in seventh grade, helping the math circle teams score among the top teams in the state each year. In February 2010, in his final attempt at the AMC12, he contributed to a record best-ever team performance by any Albany Area Math Circle on any AMC math contest--the Albany Area Math Circle AMC12B team of Dave Bieber, Andrew Ardito, and Schuyler Smith placed first in New York State. Indeed their team's performance on that remarkably challenging 2010 AMC12B contest was among the top five team scores in the country. Two younger students whom Andrew has coached in the past both earned top individual honors on honor roll for AMC12A/B: Schuyler Smith tied for high-scorer in the NY-NJ region on that contest, and Matthew Babbitt earned honors as the region's top-scoring student in ninth grade or below.

The first time Andrew qualified for the USA Math Olympiad, he had to take the 9-hour contest by himself, because he was the only qualifier in our area that year. However, he has generously shared his olympiad problem-solving skills with other students, and this has resulted in the camaraderie of a growing number of math olympians in our math circle taking the contest together in every year since then. This year's qualification process was exceptionally rigorous and yet our math circle had a record-tying five students taking the USA Math Olympiad, as Andrew was joined by Felix Sun (Shenendahoah HS junior), Matthew Babbitt (homeschool freshman), Schuyler Smith (homeschool junior), and Dave Bieber (Niskayuna HS senior).

There is an ancient metaphor invoked by many philosophers, mathematicians, and scientists that seems especially apt here: "standing on the shoulders of giants."

Andrew once stood on the shoulders of earlier members of Albany Area Math Circle, and he gives special credit to founding member and "soul of the Albany Area Math Circle" award-winner Drew Besse for inspiring and mentoring him.

In turn, Andrew's shoulders have surely provided strong support to launch many other students in our math circle, who in turn launch other students, and so on and on. Although Andrew and Dave will be heading off to college in the fall, our other olympiad veterans, who also include Ashley Cho (a rising senior at Emma Willard) and Jay White (a rising homeschool senior), as well as Felix, Matthew, and Schuyler, will surely continue to provide strong mentorship shoulders to future aspiring olympians in our math circle. Andrew has also helped to encourage and inspire a growing number of participants in the USA Physics Olympiad contests: our math circle had a record four students reach the top level of that contest series this year, with Dave, Schuyler, and Gurtej Kanwar (a Bethlehem HS junior) all joining Andrew in taking the USAPHO.

☞ Andrew has consistently risen to the occasion with outstanding and enthusiastic contributions to countless collaborative math team rounds in a variety of venues. He began participating on our teams for NYSML and the Harvard-MIT Math Tournament in middle school. PUMaC, the Princeton University Math Contest, did not start until he was a freshman, but he participated in that contest, studded with a stellar field of many of the strongest math olympians in the country, throughout his four years of high school, accruing multiple top 10 honors as a sophomore, junior, and senior years, including 2nd place rankings in combinatorics (2009) and number theory (2007). His record at ARML is also especially noteworthy: in 2007 and 2010, he made the national tiebreak rounds and received Individual High Scorer of the Meet awards, while in 2008 and 2009, he earned honors as high scorer on the Upstate NY team.

As Team Captain, his leadership contributions to team rounds at NYSML, HMMT, PUMaC, and ARML have been exceptionally impressive. Competing against extraordinarily strong teams from all over the state, the country, or even the world, he has led our math circle teams to many great rankings. Special highlights for teams he has led as captain include 3rd place rank at NYSML 2009--and "Most Improved" team award, 2nd place rank at NYSML 2010, 1st place rank for NYSML local in 2009-2010, a top 10 rank at HMMT 2010, and 3rd place Power Round at PUMaC in January 2009.

The crowning accomplishment for Albany Area Math Circle during Andrew's time as captain came in the national/international ARML Power Contest in 2009-2010. That contest is a collaborative round in which all members of the math circle were able to work together, and it's key for the captain to find ways to make the most effective use of the collective talents of rookies as well as veterans. There are two rounds to that contest each year, one in November and one in February. There is extremely challenging competition from math circles and magnet school programs all over the country and the world. The top 10 teams are honored with plaques at ARML each June. Our math circle had done well in the past, but we had never made that top 10 list before. Andrew clearly had been thinking hard about how to draw on his years of experience and knowledge of other students' strengths to do well this year. In November, the team was thrilled to take eighth place on the honors list, and in February, they were even more ecstatic to learn they'd placed second on the February round, resulting in a fourth place rank overall for the year. The photo below shows happy representatives of the Upstate NY Math team from the Albany Area Math Circle accepting the plaque at the Penn State ARML awards ceremony on behalf of all members of the Albany Area Math Circle.

As I write these words in August 2010, Andrew is at PROMYS, a summer program designed for students with extreme talent and passion for mathematics. Andrew won the ARML Scholarship to attend PROMYS in 2008 as a student, and returned as an advanced student the following year. This summer, he is a member of the instructional staff at PROMYS, working as a counselor.

Next month, Andrew will go off to Princeton, where he plans to study math. Dave Bieber will be heading to Princeton as well, with plans to study computer science. (For more about Dave, see this post.) Dave and Andrew have been the heart and soul of Albany Area Math Circle since Beth Schaffer and Drew Besse graduated in 2007.

Our math circle will miss them both, but we hope they'll come back to visit during school breaks, as Beth and Drew and many of our alumni do. And our math circle students will look forward to seeing them when the team travels to Princeton for PUMaC in November.

And, if Dave and Andrew miss our math circle, Princeton now has one too. Our loss is their gain. Well, no, not really--it is not a zero-sum game, but rather a positive-sum game. Our intersecting circles enrich us all.

And, of course, both Andrew and Dave have left our math circle with a remarkable legacy--their friendly encouraging spirits and their passionate enthusiasm for challenge and their generous willingness to share their prodigious talents and their remarkable ability to help others work well together will remain behind in the hearts and souls of all the enthusiastic younger students with whom they have worked.

## Sunday, August 1, 2010

### Pythagorean Auctions notes

Notes from my Pythagorean Auctions session at the New York Middle School Math Teacher Workshop at Bard College are available here. I will annotate them more fully later this summer.

## Wednesday, July 21, 2010

### Pythagorean auctions and other Pythagorean excitement for middle school teachers at Bard College next week!

As I said, why should students have all the fun?
Next Monday, I will be leading a session on Pythagorean auctions at the New York City Math Teachers Circle summer workshop for middle school mathteachers at Bard College.

The inspiration for running math auctions in my session came from an idea I first read about in the book Mathematical Circles (Russian Experience) by Dmitri Fomin, Sergey Genkin, and Ilia Itenberg. Further encouragement came from a very enthusiastic presentation by Anna Burago at MSRI's Great Circles 2009 conference. Anna has many years of experience with math circles, both as a student in her native St. Petersburg and now as a lead teacher for the Northwest Academy of Sciences math circle. Here is her description:
Mathematical Auction is an exciting team mathematical contest. It combines mathematics with the elements of gambling and psychological thriller. The game starts from a period of problem solving which is followed by a round of trading. The teams bid, scheme and strategize for the rights to present the solutions to the problems. Mathematical Auction is one of the favorite competitions in the Math Circles that I run.
As an economist, of course, I was intrigued by the idea of a mathematical auction, so I had to try the idea out in our math circle. Since Anna said that the auctions are especially popular with middle schoolers, we ran some math auctions at our Albany Area Math Circle middle school math circle meetings last spring--and they were indeed very well received by the students. I hope the teachers at next week's workshop will also enjoy using math auctions to explore some interesting Pythagorean mathematics.

Here are the rules for a Math Auction:

1) The auctioneer hands the teams a set of problems--about five or six problems is considered ideal. But the problems are a special kind--Russian math circle leaders call them "research problems" because they allow possibilities for partial solutions or intermediate answers which can gradually lead to a final result.

2) Teams are given a certain amount of time to work on the problems.

3) When that time is up, the auction begins. Each team starts with a certain amount of fictional currency at the outset of the auction. In honor of Pythagoras, we'll use drachmas as our fictional unit of currency in next week's workshop, so each team will have 1,000 drachmas to use in their bidding.

4) The teams then bid for the right to present a solution to each problem. The team that submits the highest bid in the initial round of the auction gets the right to present their solution. After they present their solution, the bidding reopens so that other teams who believe they can improve on that solution may do so. The team with the strongest solution to the problem wins the value of the problem. Bonus drachmas may be awarded for cool discoveries made alone the way.

The focus of the summer teacher workshop is the Pythagorean theorem, so all the problems we will be auctioning off next Monday afternoon will use that theorem in a variety of fascinating ways--many of the auction problems will connect to the presentations that will be given by other speakers later in the week.

Here is the list of workshop presentations scheduled at the Bard math teacher summer workshop next week:

Auctions of All Things Pythagorean: Spirals, Trees, Triples, Twins, Quads, Networks, and Outcastes, Mary O'Keeffe (Albany Area Math Circle)

Showing of The Theorem of Pythagoras (movie created by Tom Apostol's Project Mathematics! at Caltech with awesome animation from MacArthur "genius" prize winner Jim Blinn )

Primitive Pythagorean Triples Sheila Krilov (Hunter College High School teacher and MATHCOUNTS coach)

Heronian Triangles David Hankin (former chair of the AIME Committee and veteran teacher/math department chair Hunter College High School)

Showing of The Proof (movie about Andrew Wiles proof of Fermat's Last Theorem)

Pythagoras in Spherical and Hyperbolic Geometry, Jim & Maria Belk (Bard College faculty)

Eight Different Proofs of the Pythagorean Theorem Gary Rubinstein (Stuyvesant High School faculty)

Fermat n=4 and Some Interesting Open Problems in Number Theory Lauren Rose (Bard College faculty)

Primitive Pythagorean Triples via Unique Factorization John Cullinan (Bard College faculty)

Almost Pythagorean Triples, and Almost Isosceles Pythagorean Triples: Connecting to Pell's Equation, Japheth Wood (Bard College faculty)

All of this mathematical excitement (as well as music and film festivals) will take place on Bard's campus next week. The area near Bard, overlooking the Hudson River, is a spectacularly beautiful setting at this time of year (word has it that Chelsea Clinton will be getting married nearby shortly after the math circle teacher workshop ends!)

There may be still room for a few more math teachers to attend the workshop--and it is even possible to arrange for continuing education credit. Please pass the word along to any middle school math teachers you know who would enjoy this experience!