Monday, April 27, 2009
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: eiπ + 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
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.
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.
Albany Area Math Circle advisor
Monday, April 13, 2009
Sunday, April 12, 2009
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.
Ashley, Liz, and Eric got scores of 33 out of a possible 36, tying for 30th place in . 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 , , and Shenendahoah High School, which all made the top 30 cumulative team honors list of the .
And yet more congratulations to two of our 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
Congratulations to the following Albany Area Math Circle students who have qualified for the USA Math Olympiad(USAMO):
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 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.