Monday, April 27, 2009

Math fortune cookies

Math Circle members Dana McLaughlin and Bea Malsky came up with a very cool idea for refreshments at a recent math circle meeting: homemade fortune cookies!

They used this recipe and tucked in slips of papers on which they had written various cool math puzzles, formulas and "factaroonies" before folding up the fortune cookies. It sounds like it took a bit of trial and error to get the folding technique right, but the ones they brought in looked and tasted great. (Important tip: I understand that it's easier to work the dough-folding if you make them larger than the commercial variety.)

A good place to look for cool math factaroonies to put into cookies is Tanya Khovanova's "Number Gossip" website. Type in a number, say, 28 or 17 or 13 or 27, into the Number Gossip website and you'll get a bunch of interesting facts and properties of that number. For example, lots of people know that 28 is a perfect number, but did you know that 28 is the only even perfect number of the form x3 + 1? Or that 17 is the number of different wallpaper groups (i.e., plane symmmetry groups)? Or that three planes can cut a donut into a maximum of 13 pieces? Or that someone who is 10,000 days old is age 27?

Here are a few ideas for cool formulas suitable for math fortune cookies to get you started. I'm sure you can come up with many more with a little searching:

The Euler identity: e + 1 = 0.

Euler's polyhedral formula V + E - F = 2
(where V is the number of vertices, E is the number of edges, and F is the number of faces of any given polyhedron.)

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ....

(Or equivalently, if we use "bimal" notation instead of decimal, we can write the previous equation as 1 = .1111... in bimal--the binary equivalent of decimal.)

The golden ratio phi has many beautiful equations suitable for use in a fortune cookie. Here's of them:

phi = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + .....

There's a wonderful world of such formulas out there! And of course, you could take the approach of posing some of these beautiful formulas in an "exploratory question" form instead of as an equation statement form.

Sunday, April 26, 2009

MAA Minute Math and the AMC problem database

If you want to "Exercise your mind daily with a problem from the AMC-8, AMC-10, or AMC-12, provided by MAA's American Mathematics Competitions," visit the MAA Minute Math website.

Note that the official solutions provided are correct, but are not always the most efficient, so it's very worthwhile to think about more creative approaches of your own.

Minute math also provides interesting data on the percentage of test-takers who got each question correct. If you get the question wrong, you can take comfort in the fact that you generally have lots of company! (And bear in mind that the sample of AMC contest-takers is not a random sample of the population. The pool of students who take AMC exams includes a disproportionate number of strong and enthusiastic math students.) Whether you got the question right or wrong, it's interesting to look at the "distractor" wrong answers, which often give insights into common misconceptions.

It's important not only to understand your own blind spots, but also to recognize the blind spots of other people. This is true both in math competitions (which often have team-based components, so it's important to anticipate areas in which teammates may make mistakes), and also in the real world (perhaps the financial bubble which led to the recent meltdown would have been "pricked" sooner if more financial decision makers had considered the blind spots of other financial decision makers.)

If you want to look at the entire database of problems used on the AMC8/10/12 contests over the past decade, they are available here in a format that allows you to select particular types of problems, for example, discrete math, geometry, etc.

NYSML congratulations!

Albany Area Math Circle's A team took THIRD PLACE!!!! in a strong field of 27 teams of the top students from across New York State with a score of 205 points at Saturday's New York State Math League (NYSML) state championship meet yesterday.

To top it off, the AAMC team came closer than ever before, pointwise, to finishing even higher, since the top two scores earned by the NYC powerhouse teams were 209 and 222.

In addition to ranking third place in the A division as well as overall this year, AAMC also won a special award for "Most Improved" with a score increase of 79 points over last year's score--this was 50 points more than any other team improved at the meet! The team got 9 out of 10 on the team round (tied for first place), a very impressive 40 on the power round (tied for third place), and did better than ever on the relays, with scores of 20 on the first one and 16 on the second one.

The individual score total of 84 was extremely impressive, tying for second with AAST's out of state (and therefore unofficially unranked) NJ Powerhouse team and only 3 points behind the top NYC team. Given that NYC had 18 USAMO qualifiers this year and AAST had 11 USAMO qualifiers this year, it is extremely impressive that AAMC racked up such an impressive total.

The first time Albany Area Math Circle participated in NYSML, in 2002, we were just getting started, and it was extremely exciting for the seven AAMC students to join with members of the Niskayuna high math club to form a composite team which wound up doing better than anyone had dreamed, and taking first place in division B. That was a HUGE accomplishment, but we didn't even dare dream back then of coming close to the unbeatable top NYC powerhouse teams in division A.

It is some measure of how far we have come that we now consistently compete in division A, and those powerhouse teams are within our sights! Our score this year was 60 points above the division B champion, and other coaches in the A division view us as a "force to be reckoned with!"

To make matters even more exciting, we brought along a record-size contingent including a number of extraordinarily promising middle school students on our B team, who energetically took on a difficult challenge and did very creditably, showing great promise for the future, especially for next year when Albany Area Math Circle will be the "home team" for the meet. Kudos to our talented, brave, and energetic middle school students and to the student coaches who have mentored, encouraged, and supported them.

It was exciting to be able to bring two full teams plus a very strong group of alternate students to NYSML this year. Again, this is a measure of how far we have come. In our very first year, it was a struggle to put together a single team of 15 students (even though the competition was held in Niskayuna that year! The problems were just too scary and it was hard to find 15 students in the Capital District who were willing to spend a Saturday working on such hard problems.)

Students on AAMC A team were: Andrew Ardito (heeg), Matthew Babbitt(heeg), Dave Bieber (Niskayuna High School), Ashley Cho (Emma Willard School), Peixuan Guo (Bethlehem High School), Gurtej Kanwar (Bethlehem High School), Paul Rapoport (Albany Academy), Markus Salasoo (Niskayuna High School), Liz Simon (Guilderland High School), Schuyler Smith (heeg), Wyatt Smith (heeg), Felix Sun (Shenendahoah High School), Eric Wang (Shenendahoah), Yipu Wang (Guilderland High School), and Jay White (heeg). Our alternates were Deepak Aron (Niskayuna High School), Adam Parower (Shaker), Brady Pelkey (Hudson Falls High School), Kyungduk Rho (Guilderland High School), and Ved Tanavde (Guilderland High School). Our AAMC B team had 10 outstanding and brave middle school students along with five high school student coach participants. The middle school student members of AAMC B team were: Cecilia Holodak (Van Antwerp), Mandy Kettell (heeg), Preston Law (heeg), Isaac Malsky (Farnsworth), Zubin Mukerjee (Farnsworth), Jien Ogawa (heeg), Elizabeth Parizh (Iroquois), Gili Rusak (Loudonville Elementary), Aniket Tolpadi (Iroquois), and Troy Wang (Acadia.) Student coaches participating on the AAMC B team along with these very promising middle school students were Bea Malsky, Dana McLaughlin, and Noah Rubin from Guilderland High School, Anagha Tolpadi from Niskayuna High School, and Lindsay White from heeg. Many other math circle students have also been coaching and mentoring the students in our middle school math circles and/or on their local MATHCOUNTS teams at local middle schools--there is great promise and enthusiasm for the future.

The high scorer on the A team was Matthew Babbitt. The high scorer on the B team was Zubin Mukerjee. Adam Parower was the high scorer on the alternate 2 team.

Thanks also to all the math circle parents who made it possible for Albany Area Math Circle to participate this year--by driving carpools, helping with scoring and proctoring, and observing the competition to help us prepare for next year--as well as for everything you do to support regular math circle meetings. Thanks especially to Mr. Babbitt for organizing so many essential details to make sure that the expedition to NYSML was a success again this year! Thanks as well to Professor Krishnamoorthy for all his excellent work in helping our students prepare mathematically.

I very much appreciate all the offers of help from parents that I have received about preparing for next year, when Albany Area Math Circle will be the host team.

Congratulations to everyone who participated in and contributed to such an amazing community.

Mary O'Keeffe
Albany Area Math Circle advisor

Monday, April 13, 2009

Hudson River Undergraduate Math Conference to be held at Union College Saturday April 18

This coming Saturday, April 18, students and faculty from colleges and universities all over upstate New York and western New England will be coming to Union College in Schenectady to present a dazzling smorgasbord of talks about a variety of mathematical topics at the sixteenth annual Hudson River Undergraduate Math Conference (HRUMC).

Students from Albany Area Math Circle who have attended a previous HRUMC have found many of the talks fascinating and very understandable. Most of the presenters are undergraduate math majors, and all the level 1 talks are generally very accessible to advanced high school students who enjoy challenging mathematics of the sort we do in our math circle. In fact, if you have an idea for giving such a talk yourself, you might be inspired to submit a proposal to next year's HRUMC. The organizers have accepted some such proposals, and I can recall at least one year in which a high school student's presentation was extremely well-attended and well-received by the faculty and students in the audience. If you think you might like to present at this conference in a later year, you should definitely attend this year.

Most of the talks are quite short, so you have the opportunity to sample lots of different math topics in the one-day, both pure and applied. About 20% of the talks are by faculty and 80% by students. Here's just a selection of a few of the topics that struck my fancy.

The keynote speaker is Professor Erica Flapan from Pomona College, who will give a talk called "When Topology Meets Chemistry," about the connections between knot theory and molecular chemistry. (If this subject interests you, you might enjoy Martin Gardner's fascinating book on related topics, The New Ambidextrous Universe, which provides a highly engaging introduction to some of the key concepts.)

Russell Sage senior Sybil Wojcio looks at Lewis Carroll's mathematical recreation puzzles and guessing games. (Yes, THAT Lewis Carroll! The author of the Alice books was a mathematical logician at Oxford University. If you want to know more, read Martin Gardner's Annotated Alice and/or Gardner's edited collection of Lewis Carroll's mathematical recreations, The Universe in a Handkerchief.)

Westfield State juniors Jamie Cocomazzi and Sarah Grella will talk about their work using ChordGeometry software to investigate the mathematics of music, explaining why the space of triads looks like a triangular prism while the space of 2-chords is a Moebius strip. Prof. Larry Knop from Hamilton College will talk about some of the mathematics underlying Google's page-rank algorithm. Williams College junior Christophe Dorsey-Guillaumin will give a talk for advanced undergraduates about the Riemann Hypothesis and quantum mechanics. Union College Prof. Bill Zwicker and senior math major Andy Mackenzie will present a two-part session on applications of game theory to the design of voting machines. Prof. Zwicker is a noted expert on the mathematics of voting theory as well as a great teacher. He was also one of the original founders of HRUMC, along with Prof. Colin Adams of Williams College, Prof. Emelie Kenney of Siena, and Prof. David Vella of Skidmore. HRUMC has been an enormously successful program which has inspired the launching of a number of other similar conferences around the country.

The talks listed above are just the tip of the iceberg--there's much more to choose from. The complete list of talks, presenters, and downloadable abstracts for each talk is available here. Note that each presentation has a "Topic Level" rating of 1 or 2. Albany Area Math Circle students who attended HRUMC in previous years have found all the material in level 1 sessions very accessible and well within their mathematical comfort level. Those AAMC students who had already done advanced college-level work in abstract algebra, number theory, analysis, and/or topology also found level 2 sessions quite accessible as well.

Albany Area Math Circle members who would like to attend should email Advisor Mary O'Keeffe ASAP.

Sunday, April 12, 2009

Congratulations to Felix

The April 2009 issue of Math Horizons credits "Felix Sun of the Albany Area Math Circle" with submitting a correct solution to a problem posed in their November 2008 issue.  Although the problem was also successfully solved by three other submitters, the article made clear that Felix' submission stood out from the others because his solution took care to establish that the answer did not depend on a possibly unrealistic assumption.

Math Horizons is a publication of the Mathematical  Association of America which features "problems for students at the undergraduate and (challenging) high school levels" in each issue. Some other math journals with problem sections include:  the American Mathematical Monthly, Mathematics Magazine, the College Mathematics Journal, Crux Mathematicorum, the Harvard College Math Review, and the Pi Mu Epsilon Journal.  All these journals encourage students to submit solutions to the problems and will publish the names and affiliations of those who submit correct solutions.   Occasionally, the journals will also publish commentary highlighting particularly noteworthy aspects of a student's submission, as Math Horizons did with Felix' solution.     

Local college and university libraries subscribe to most of these publications, and most are also available at special discounted student subsciption rates.   Working on solving and writing up the problems in them is a great way to develop your skills in mathematical expository writing, as well as in perseverence and problem-solving, since these problems typically take a lot more time and effort to solve than typical contest problems.  

Kudos to Felix!

New York Math League congratulations

Congratulations to Albany Area Math Circle members David Bieber, Ashley Cho, Liz Simon, Eric Wang, and Yipu Wang, who all made the top cumulative individual student honors list in the New York Math League High School division.  

Ashley, Liz, and Eric got scores of 33 out of a possible 36, tying for 30th place in New York State.   Dave and Yipu were two out of only seven students in the state who got perfect cumulative scores.

Congratulations also to the entire teams of participating students at Emma Willard School, Guilderland High School, and Shenendahoah High School, which all made the top 30 cumulative team honors list of the New York High School division.    

And yet more congratulations to two of our Middle School Math Circle members, Cecilia Holodak and Aniket Tolpadi, who made the top statewide individual honors list for the eighth grade division of the New York Math League.  Cecilia was tied for 22nd student in the state and Aniket for 9th place student in the state.  This is all the more impressive given that both students are seventh graders competing in the eighth grade division of the league.  Congratulations also to all members of their school teams at Van Antwerp and Iroquois respectively, which both ranked on the school team statewide honors list as well.

Friday, April 10, 2009

USAMO Congratulations!!

Congratulations to the following Albany Area Math Circle students who have qualified for the USA Math Olympiad(USAMO):

Andrew Ardito
Matthew Babbitt
Schuyler Smith
Yipu Wang
Jay White

According to USAMO website:

The USAMO (United States of America Mathematics Olympiad) provides a means of identifying and encouraging the most creative secondary mathematics students in the country. It serves to indicate the talent of those who may become leaders in the mathematical sciences of the next generation. The USAMO is part of a worldwide system of national mathematics competitions, a movement in which both educators and research mathematicians are engaged in recognizing and celebrating the imagination and resourcefulness of our youth.

The USAMO is a nine-hour two-day proof-based contest with six difficult problems that would "challenge most professional mathematicians," according to the Mathematical Association of America, the professional society which sponsors it.    Each year, hundreds of thousands of the strongest high school math students in the country take the challenging AMC preliminary series of exams and only the top 500-scoring students on those exams qualify for invitations to write the USAMO.  It is remarkable that five Albany Area Math Circle students are among the 500 this year--our qualifying students constitute 1% of the top-scoring students in the USA and Canada this year, even though the Albany area surely constitutes much less than 1% of the total population of the US and Canada.

According to the USAMO website:

The twelve top scoring USAMO students are invited to a two day Olympiad Awards Ceremony in Washington, DC sponsored by the MAA, the Akamai Foundation, the Microsoft Corporation and the Matilda Wilson Foundation. Six of these twelve students will comprise the United States team that competes in the International Mathematical Olympiad (IMO). The IMO began in 1959; the USA has participated since 1974.

The USAMO will take place on Tuesday April 28 and Wednesday April 29.  Congratulations and best of luck!