AMC8 coming Tuesday!

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

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

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

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

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

A few things to remember:

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

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

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

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

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

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

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

AMC8 Tuesday Nov 17

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

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

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

Great short math videos

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

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

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

A puzzle celebrating Martin Gardner's 95th birthday

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

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

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

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

He quit his job with Humpty Dumpty.

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

How is this possible?

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

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

As the article notes:

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

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

Congratulations on AMC12 Team Standings!

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



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

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


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

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

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

2009 AMC10/12 Honors and Award Summary Book

2008 AMC10/12 Honors and Award Summary Book

2007 AMC10/12 Honors and Award Summary Book

2006 AMC10/12 Honors and Award Summary Book

2005 AMC10/12 Honors and Award Summary Book

2004 AMC10/12 Honors and Award Summary Book

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

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

Good luck and best wishes to the new math circle starting up in Princeton NJ

A new math circle for middle school and high schools students is starting up today in Princeton, New Jersey! The legendary Professor John Conway is going to be giving a talk and Albany Area Math Circle founding member Alison Miller, now a math grad student at Princeton, will lead a problem session afterwards.

AAMC high school students who are leading middle school math circles may not be able to get John Conway to appear in person, but that doesn't mean you can't introduce the mathematical excitement of his work to your students.

Here's a video of John Conway talking about his work on cellular automata and The Game of Life:



If you don't know about his work on cellular automata and The Game of Life, you and your students are in for a treat. Visit this link to learn more about the concept and play an interactive version of the game.

Understanding the real world policy applications:

Conway's Game of Life relates very nicely to some extremely influential work done by economist Tom Schelling, who won a Nobel Prize for his work applying game theory to real life. Schelling's books are extremely well-written, lucid and thought-provoking. They don't require a lot of background. Micromotives and Macrobehavior is a good place to start. Schelling started working on models of racial segregation patterns in the 1950s and 1960s by creating simple cellular automata rules with pennies on a chessboard. Another great Schelling book has a title that I love and can (deeply!) relate to: Choices and Consequences: Perspectives of an Errant Economist. Here's a video from economist Tim Harford briefly presenting and discussing Schelling's model, using eggs to demonstrate cellular automata:



To explore the math more deeply:

Check out the links on this page at Wolfram demonstrations. The first page allows you to change parameters and visualize different ways that his Game of Life will play out, but make sure to check out the links on the right side of the page as well.

John Conway is co-author, along with Richard Guy and John Berlekamp, of a great series of books on mathematical game theory, called Winning Ways for Your Mathematical Plays. It's in the fun but deep and sometimes hard category--highly recommended for those who want to delve more deeply into this subject. Google Books also offers a limited preview of volume 2 of the series which I've linked below. If you follow the google books link, you can find what local libraries near you own the book.