News from the Math Wizard

I'm moving my blog to www.japheth.org

2011-09-17T08:04:00.003-04:00

Dear Readers,

Over the last few years I've been exploring how to run my own blog and website. Now with the examples and encouragement of many friends who have really inspirational blogs (like Research in Practice and the Mathbabe) as well many more blogging acquaintances and others who tweet, post amazing things on Facebook and Google+, I've decided to move my News from the Math Wizard posts over to my website at www.japheth.org.

I will be using my webpage for several purposes and several audiences: continued news from the math wizard, as well as posts related to my work in mathematics and in education. And perhaps more.

I hope you will follow me to www.japheth.org!

Teachers' Math Circle Talk at Math Fest

2011-07-04T10:45:00.003-04:00

Photo Credit: Rusty Rogers

I'm excited to announce that I'll present the demonstration Teachers' Math Circle Class at MathFest!

For those not in the know, MathFest is the biggest and best national summer math conference, organized by the Math Association of America (MAA). There will be many activities sponsored by the Special Interest Group on Math Circles for Students and Teachers (SIGMAA-MCST) including two contributed papers sessions, a math wrangle and demo classes for students and teachers.

My demonstration class will be my talk on Nim and Jim, which I blogged about here in April. Every time I presented this activity about impartial games it gets better and more interactive. So if you've seen it before, you'll have a lot of fun, but if you've never seen it, then you're guaranteed to have a great time!

MathFest this year is August 4-6 in Lexington, Kentucky.

The Math Babe and the Science Babe

2011-06-26T15:26:00.001-04:00

Two interesting blogs that I follow are called, respectively, The Science Babe and mathbabe. No, neither of these blogs are prurient websites that are weak on content. Instead, they both seek to reclaim the terms from websites that are.

The Math Babe is none other than Cathy O'Neil, a Ph.D. Number Theorist who left academia about four years ago for industry. She blogs not only about being a female mathematician, but also the mathematical techniques and tools that she is acquiring right now in industry. I do hope that she blogs about Number Theory too, although it's exciting to read about how she is mastering Python and R.

A recent post that I find interesting is about Working with Larry Summers. Summers was the president of Harvard University between 2001 and 2006, who resigned in disgrace after the bad publicity generated in part by his comments about women's aptitude in mathematics. After this, he worked at D.E. Shaw, which is where Cathy got to work with him. His project there was apparently to chase dumb-money. Any profit involved was at the cost to pension funds that many of us hope to retire on. Yuck!

Now, how about the Science Babe? Let's save that for another post!

Always be Prepared

2011-06-11T14:37:00.000-04:00

Back in April, I wrote about Math Circles and Safety, and a large part was inspired by Brandy Wieger's presentation at the Math Circles on the Road 2011 conference in Houston. You can now watch her talk here:

This is one of the videos recently posted by The National Association of Math Circles. Check out their YouTube channel here: MathCircles.

Do Not Litter And Feed The Birds

2011-05-27T08:32:00.001-04:00

A sentence in natural language can often be interpreted in two ways, but in mathematics, we must be precise in our use of language and avoid ambiguity. This often least to humorous interpretations of street signs and the results can be adorable.

Take this street sign example, which I photographed in September of 2010 at the Happy Paws pet store near NYU. It's a wonderful sign on a messy street that immediately attracted my attention and confusion.

Do Not Litter And Feed The Birds

Please leave a comment below on how you understand this sign:

A) Two separate commands: you should not litter AND you should feed the birds. Why?

B) You should neither litter nor feed the birds. Why?

C) You should either not litter or not feed the birds, or do neither. Why?

D) All of the above (and why?)

E) Explain a distinct interpretation.

Magic Cubes

2011-05-20T21:59:00.000-04:00

Last weekend was the last meeting this semester of the Bard Math Circle at the Kingston Library, and we made magic cubes. A magic cube is a large cube made up of eight smaller cubes that are hinged together in such a way that they rotate through themselves magically, revealing many surfaces. This activity is a little more involved that what we normally do, so you either have to follow directions, or problem solve.

Several museums sell these with artistic pictures on several surfaces, but you can make your own. It's more fun!

Instructables has instructions that also use wooden cubes, but go a step further with photos: http://www.instructables.com/id/Crazy-Foto-Cube/. The downside of Instructables is that they want you to become a member.

Here's a site, Chica and Jo, that gives full instructions in a very nice presentation: http://www.chicaandjo.com/2008/05/08/magic-folding-wooden-photo-cubes/

A company that manufactures magic cubes as a promotional product. Here's one of their simpler models, but they have quite a few to choose from:
http://www.magicubes.com/product/promotional-product-puzzle-for-trade-show-events-magic-cube-gifts/45

The German website http://www.mathematische-basteleien.de/magiccube.htm also has some clear instructions. Their instructions are revised for the clearest instructions yet: http://sofia.nmsu.edu/~breakingaway/Lessons/MFC/MFC.html.

Enjoy!

The Job Market is HOT this week!

2011-05-09T22:43:00.000-04:00

The job market in NYC for teachers and schools has reached a new level of irrationality this year with unreasonable threats of a massive teacher layoff (yet again), and a seemingly eternal (yet porous) hiring freeze that separates a pool of talented but anxiety-ridden teaching applicants from ever more desperate public school principals in need of qualified instructors.

My prediction is that the big winner this year is the charter school. This is not a big surprise; their lack of restriction in hiring gives them unfettered and early access to the candidate pool. I've also heard that they have a direct line to bottomless pits of hedge fund monies.

A lack of restriction and unlimited funding is one thing, but last week I saw an unstoppable weapon, the charter school recruiter!

I met my first recruiter last week while observing one of my MAT math students at one of my favorite urban high schools on the Lower East Side of Manhattan. My student was doing a brilliant job: he was motivating, provided excellent scaffolding and carried out a wonderfully planned lesson. Several times throughout the lesson, his students had truly become thinkers and mathematicians.

And then, a stranger appeared. In walked an elegant and confident young lady. She was dressed for success and exuded expertise. In the clearest pantomime performance I've seen, she asked my student teacher, "can I videotape your lesson with my flip video recorder?" Then she sat down next to me in the back of the room, and I introduced myself.

It turns out that she is one of two full-time recruiters working for a cluster of four charter schools in Harlem. She was following up on my student's interview with a visit to his classroom, just long enough to let him know that they were very interested, and to record some footage of his lesson for the charter school principal.

Even in this market, where I expect many of my amazing students to linger on the job market through the summer, and to be excited when any tenuous job offer comes in (after the first day of the school year, and with a long commute), I knew that this student was not going to be on the job market for long.

The best teaching candidates never stay on the market, even in the worst of times. (I suspect that the very best candidates never even step foot on the job market.) But there are exceedingly good teachers on the market right now. This week, the job market is HOT!

Last year an extremely good math candidate contacted me for help finding a job. I sent out emails, made phone calls on her behalf, and offered her encouragement (there was a lot of anxiety even in last year's market). Five days later, she landed her first interview on a Friday. The school immediately invited her in for a demo lesson on Tuesday and offered her the job on the spot. She wrote to me that "Surprisingly, Discovery HS took three days to carry out the hiring process- interview, demo and decision making." In all, she was on the market for only 7 weekdays.

How can we compete with professional recruiters, and schools that can turn around a hiring decision in just three days? It may seem simplistic or silly, but I think it is easy:

Confirm receipt. When a prospective hire applies, they should get a personal response promptly. This immediately reflects that they will be treated with respect, from the very beginning.

Explain the process. The worst part about being on the job market, is the experience of limbo. It takes almost no effort to explain how the hiring process will work, and approximately how long each part will take.

Check completeness of the application. Let them know if there is anything missing, and if so, what it is.

Interview them if they are of interest.

Let them go if they're not. Don't keep their hopes up, but say it nicely.

Inform them what comes next. How long will it take? How close are they to a hiring decision?

Answer them promptly. Don't let a decision linger!

If you're curious how I know that this method works, just check out what the first initials spell. This is exactly why the MAT program brings in students year after year, and is able to compete with the more established schools, as well as the cheaper programs.

Happy Hiring!

Math Circles and Safety

2011-04-24T17:17:00.002-04:00

The Bard Math Circle is growing rapidly this year. Perhaps this started because the wonderful Kingston Library director, Margie Menard, sent out a press release that was picked up by the local media, or that we've worked hard to develop a consistent and predictable schedule. But the fact is that the Bard Math Circle has found its niche: libraries.

This has me thinking of ways to ensure that our activities are safe spaces.

Safe to Take Risks
One important aspect of this is that participants should feel safe to take mathematical risks. Considering all the adults I meet whom have experienced some sort of mathematical trauma when they were young, this is crucial. Participants need to feel safe to explore mathematical ideas.

Here's a quote from a Kingston grandmother who brought her granddaughter to yesterday's math circle. After hearing this, I know that we're doing a great job:

Since she's been coming here, her math has improved. She thinks about things now. This is the most worthwhile thing she's been involved in.

Right on! We're definitely going to come back to this topic in the future.

Physical Safety
Another important aspect of safety for our Math Circles is physical safety. I traveled to Houston last month to attend the Circle on the Road Spring 2011 conference. Besides some incredible mathematics, the most interesting presentation, by far, was Brandy Wiegers' talk, Always Be Prepared. Brandy has been involved in Girl Scouting for over 20 years, and as a result, she is the mathematician I'd most want to be with in an emergency.

Here's Brandy's abstract:

Math Circles should be fun and engaging. To keep it this way it is important to be prepared with a box of tricks and some quick plans to ensure safety. In this session we'll discuss what we keep in our Math Circle Box of Supplies, important legal aspects of working with minors including adult to student ratios and the buddy system. We'll conclude the session discussing participant waivers and plans for emergencies. With a little bit of work we can all be more prepared to ensure that we never need to use our emergency plan. This way we can get back to math and everyone can have fun!

Brandy's talk was really a Math Circle milestone. Talking about safety means that Math Circles are established, and that it's now time to plan for safety. What procedures and guidelines does your Math Circle have in place to ensure the physical safety of your students? I think this is the beginning of a Math Circle discussion at the national level.

Well, Brandy did give us a lot of useful information. Perhaps the best thing she did was point us to the Girl Scouts, who've put a whole lot of thought into how to keep girls safe.

Safety-Wise
The decades-old Girl Scout safety bible is a publication known as Safety-wise. I just found out that this publication has been replaced with the new publications:

Volunteer Essentials
Safety Activity Checkpoints
Risk Management

Here are links to some specific online Girl Scout safety pages:

These safety guidelines are focused on keeping girls safe, but one thing I know for sure - if it's good enough to keep the girl scouts safe, then it will keep all of our math circle students safe!

Now, let's get back to math and have fun!

Jim and Nim

2011-04-22T11:54:00.003-04:00

I presented my talk "Jim and Nim" yesterday at the Bard Math Seminar, and managed to fill the house.

My introduction actually extended over two days, as I visited math and CS classes on Wednesday and Thursday and personally invited students to my talk. This was really fun, and I got to meet some of Bard's amazing undergraduates. I teased them with the Magic Birthday Trick:

Have you seen this?

Identify the boxes that contain the day of the month that you were born. (For example, I was born on the 18th, and the number 18 appears only in the two boxes on the right.)
Add up the numbers in the top left of each box that includes your birthdate. (For me, that's 2 + 16.)
The sum is mathemagically your birthdate.
Share in the amazement!

Here's a picture that I used to explain why this trick works. I'll leave it to you to puzzle it out!

Another teaser I shared is the game I call 21-Nim. Start with 21, and on your turn you subtract 1, 2 or 3. The first person to reach 0 is the winner. (My MAT student Kristen used this at our Pi Day Celebration for the students at 345 Brook Avenue in the Bronx.) What's interesting here is the set of losing numbers, and I challenged the Bard students to find them. In a losing position, the next player will lose, if her opponent plays correctly. Can you find the losing positions?

We didn't play 21-Nim during my talk, but we did play the classic game of Nim, with Teddy Bears:

Here's a picture of me, getting ready to hand out the teddy bears.

(This amazing photo is by Rusty, who was visiting Bard with his son Kyle the day of my talk.)

The Game of Nim

This is the real game of Nim, whose winning strategy was ﬁrst described and proved in a math research paper published in 1901. The game starts with several piles of counters. Players alternate moves: on your turn, select a pile and remove at least one counter from that pile. You win if you take the last counter from the last pile. That is, the last player with a legal move wins. If you know the set of losing positions, then you know the winning strategy!

Strategy

I had the participants play a few games of Nim, just to get the feel of the game and to start identifying the losing and winning positions. The amazing thing about impartial combinatorial games is that in each game position (with a finiteness condition) there must either be a winning strategy for you, or your opponent has a winning strategy. I call these two possibilities Winning and Losing positions (the standard notation from Winning Ways is N and P positions), which brings up a natural opportunity to use quantifiers:

If you are in a Losing position, then EVERY move you make leads to a Winning position for your opponent.
If you are in a Winning position, then there EXISTS a move that leaves your opponent in a Losing position.

I represented this in the following diagram:

Jim

Did I mention that I invented a Nim-type game that I call "Japheth's Nim", or Jim for short? One day in February 2011, I was running around Prospect Park in Brooklyn and thinking about how to motivate the strategy for Nim. Halfway around the park, I realized that a visual representation of the binary number strategy could be described independently of its connection to binary numbers and exclusive OR. I used the rest of my run to figure out just how to describe a Nim move visually in binary, which resulted in the game of Jim.

A Jim game starts with several rows of red and yellow tokens. Players alternate moves: select a row, and change one or more tokens (yellow to red or red to yellow). The ﬁrst token to be changed (from the left) must be yellow (but does not need to be the leftmost yellow token). The last player with a legal move wins. Equivalently, if you only see red tokens, then you’ve just lost!

Here's a 3-row Jim game:

Can you explain why this is a Losing position?
One of the most exciting parts of the talk was at the end, when I revealed the connection between Jim and Nim with this slide:

Well, you might have had to be at my talk for this to make sense. Please leave a comment below to describe how Nim and Jim are related!

Hungarian Dancing and Sorting Algorithms

2011-04-21T07:55:00.000-04:00

My brother and later a colleague sent me this link to these videos demonstrating Computer Sorting Algorithms through Hungarian Folk Dance:

The Bubble Sort and Shell Sort algorithms are demonstrated. Wow!

Credits from the video:

Created at Sapientia University, Tirgu Mures (Marosvásárhely), Romania.Directed by Kátai Zoltán and Tóth László. In cooperation with "Maros Művészegyüttes", Tirgu Mures (Marosvásárhely), Romania.Choreographer: Füzesi Albert. Video: Lőrinc Lajos, Körmöcki Zoltán. Supported by "Szülőföld Alap", MITIS (NGO) and evoline company.

So, this is actually a Transylvanian video.

I first learned the algorithms in high school from the Art of Computer Programming, ancient tomes written by Donald Knuth, and also in my AP Computer Science class, the first year it was offered. We wrote our code on paper, and eventually implemented the programs in Pascal on state-of-the-art Apple ]['s, once the teacher figured out how to load it on the computers!

Fiber Arts and Mathematics

2011-04-20T20:10:00.002-04:00

Wow! This book claims to show 8 ways to knit a Sierpinski's Triangle!

Crafting by Concepts: Fiber Arts and Mathematics

by sarah-marie belcastro and Carolyn Yackel (editors).

The publisher, A K Peters , has one of the most amazing math catalogs that I know of. Very much worth browsing!

How to Celebrate Pi Day

2011-03-31T21:11:00.002-04:00

The MAA now hosts a Pi Day webpage: http://maa.org/piday/
This is a must-visit site for next year's Pi Day.
How did you celebrate Pi Day?

Bard Math Circle in the News (bring your kids!)

2011-01-17T10:48:00.002-05:00

Dear Friends,

The Bard Math Circle made the front page of today's Daily Freeman with a story about our math outreach activities at the Kingston Library. COUNTING ON FUN: Math Circle at Kingston Library takes middle school kids beyond boring basics (video) (Monday, January 17, 2011) http://dailyfreeman.com/articles/2011/01/17/news/doc4d33674c5f430298712442.txt

The Bard Math Circle meets twice per month at the Kingston Library with a program of math and logic games, problem solving and hands-on math activities, designed for middle school math students. Everyone is welcome! The next meeting will be on Saturday, February 12, 1-3pm.

The Bard Math Circle was previously featured here:

Las Noticias: Por el amor a las matemáticas (Wednesday, October 06, 2010)

http://www.lasnoticiasny.com/articles/2010/10/06/comunidad/doc4cacee4f1d7bd487053172.txt

and in our La Voz: Círculo de matemáticas en Kingston (Agosto 2010)

http://lavoz.bard.edu/archivo/archivo.php?id=10568

What is the Bard Math Circle?

The Bard Math Circle, which targets middle school and elementary school students in the Mid-Hudson Valley, was formed in 2007 by Bard College Professors Lauren Rose and Japheth Wood. The Bard Math Circle is run jointly by students, under Bard's Trustee Leader Scholar Program (http://inside.bard.edu/tls/), and math faculty at Bard College. Student leaders have included Shelley Stahl, Ezra Winston, Elias Halloran and currently Jackie Stone. Outreach activities have focused on Kingston and Tivoli.

In the summer of 2010, Bard hosted the New York Math Circle's Summer Workshop 2010 for teachers (http://nymathcircle.org/2010workshop). I brought (primarily NYC) middle school math teachers to Bard for a week-long residential math immersion experience. The workshop sessions were led by Bard math professors, and instructors from the New York Math Circle and the Albany Area Math Circle. The 2011 workshop, which is scheduled for the week of July 25-29, will welcome both middle school and high school math teachers to Bard.

The New York Math Circle was featured recently in the news on NY1, "NYC's 24-hour Newschannel on the web":

Bonus Curriculum Is Greatest Common Factor For Local Math Masters (12/26/2010)

http://manhattan.ny1.com/content/top_stories/131224/bonus-curriculum-is-greatest-common-factor-for-local-math-masters?ap=1&MP4&r=4320001457

Please let me know if you would like to participate in any of these activities!

Thanks,

Japheth Wood, PhD

Mathematics Faculty

Bard MAT Program

--
"If you can't solve a problem, then there is an easier problem you can solve: find it."
-George Polya

Roundtables at East Side Community High School

2011-01-06T22:21:00.000-05:00

I just got word from Tom Mullen, the AP at East Side Community High School, that the roundtables are back! Here's the link to sign up:
https://spreadsheets.google.com/viewform?formkey=dFRqbzFTZzI5OWtGeGY1eDFlN0NqQWc6MA

Clicking on the above link brings you to the description and schedule below. I've been attending the round tables for several years, and they are definitely worth attending.

East Side Community School's January 2011 Roundtable Presentations

Thank you for your interest in participating in East Side Community School's Portfolio Roundtable Presentations. These 6th through 12th grade presentations are an authentic and powerful way for our students to demonstrate what they have learned this year. Your participation makes the experience even more meaningful and rewarding for our students.  Roundtables run from Friday, January 21 to Friday, January 28.  If you would like to participate, please complete the form below.  You will get to choose the date, subject, grade level, and the time of the presentation.  Feel free to sign up for as many presentations as you like.  If you have colleagues who would like to attend, please have them complete the form or you may complete the form for them.   Once you complete the form, you can assume that you are confirmed for that time slot (no confirmation email will be sent).  East Side Community High School is located on 11th St. between 1st Ave. and Ave. A.   You may go to the main office on the second floor to find out your room number.  Please arrive a few minutes before the presentations begin.    The following presentations take place on the days below: 



Friday, 1/21:           9th Physics; 10th History; Algebra 2; 12th English 
Monday, 1/24:        Algebra; 10th English; 11th Biology; 12th History 
Tuesday, 1/25:       11th English; 10th Chemistry; Pre-Calculus; Calculus, 9th               History; 7th English; AP English 
Wednesday, 1/26:  9th English; Geometry; 8th Science; 6th Science; 7th History 
Thursday, 1/27:      6th English; 12th Science PBATs; 8th Math; 7th Math; 11th History 
Friday, 1/28:           6th Math; 8th English; 7th Science; 8th History    


If you would like to participate complete the form below.  If you have any questions, contact Tom Mullen, Assistant Principal, at tomm@eschs.org or (212) 460-8467 x3152.

NY1 News Article about the New York Math Circle

2011-01-02T12:45:00.001-05:00

The New York Math Circle was recently featured on NY1 ("NYC's 24-hour newschannel on the web"). Check out the video here:

Bonus Curriculum Is Greatest Common Factor For Local Math Masters

Many thanks to TV reporter Shazia Khan for pursuing this topic! She got in touch with the New York Math Circle (NYMC) and I helped coordinate the filming. What came out in the video was filmed in late October, and shows an amalgam of the middle school and high school classes that day.
I learned a lot about how much effort the parents make to get their children to these high-quality math enrichment classes. Some devote their entire Saturdays to shuttling their children from activity to activity, and see it as an investment in the future.
There was one notable exception that truly impressed me - one of the middle school students takes the subway train all the way from the Bronx for class on his own. He was clearly having a great time at the class, and it completely shows in his interview. Can you spot him in the video?
The TV news reporter, Shazia, also had a lot of fun visiting the New York Math Circle, and is interested in covering further stories in STEM (Science, Technology, Engineering and Mathematics) Eduction. You can see more of her videos here: Shazia Khan news clips
If you know of a good story for Shazia, please leave a comment, and I'll make sure your idea gets passed on!

Canada/USA Mathcamp article

2010-12-31T17:27:00.000-05:00

Here's a wonderful article I just read about (Canada/USA) Mathcamp:

Finding Nirvana in Numbers

https://simonsfoundation.org/mathematics-physical-sciences/news/-/asset_publisher/bo1E/content/finding-nirvana-in-numbers?redirect=/mathematics-physical-sciences/news

Mathcamp is a summer math enrichment program for high school students that boasts an incredible alumni base of mathematicians, whose instructors are also something of a dream team. They move campuses from summer to summer, and are able to roll out each summer's program flawlessly. The program seems to be so successful, that I suspect it is now run completely by alums.

One wonderful thing about how Mathcamp is organized, is that this flexibility makes it replicable to some extent. Take for example Mathpath, a similar program, but for middle school students. Also, the Summer Program in Mathematical Problem Solving (SPMPS) for underserved middle school students. In some sense, both these programs "spawned" off of Mathcamp.

In high school, I attended a similar program: The Hampshire College Summer Studies in Mathematics or HCSSiM. The Hampshire program, run by Hampshire College math professor David Kelly, has raised generations of mathematicians and also has an incredible group of alumni. Honestly, you can't swing a dead cat at a math conference without hitting a HCSSiM alum. (Please don't bring your dead cat to the next math conference. Bring a yellow pig.)

The HCSSiM program seems less replicable, but that hasn't stopped alums who are recently getting started in the growing Math Circle movement. Take, for example, the New York Math Circle (NYMC), which has programming for high school and middle school students (and also teachers; more information on that soon!) and in Chicago, the soon-to-arrive Payton Math Circle. Another fairly recent math enrichment program started by an HCSSiM alum is the Math Prize for Girls contest. The Math Prize for Girls is working hard to encourage girls to achieve at the highest levels by offering incredible cash prizes. Check out their new blog! These efforts all involve HCSSiM alumni.

Out of curiosity, if you know of other math enrichment groups run by HCSSiM alums, please leave me a comment.

All of these math enrichment opportunities are wonderful, but still out of reach for most students who show interest, motivation and potential in mathematics. The Math Prize for Girls is a step in the right direction, and an even more ambitious project is the Summer Program in Mathematical Problem Solving mentioned above.

How can the math community provide enrichment opportunities to underserved students who are just out of the loop? (Please leave me a comment if you have an idea). Is the notion of mathematical talent, so often mentioned, actually damaging, as my friend Ben Blum-Smith suggests in his blog? How could mathematical resources best be used to offer high quality math enrichment to interested students? What about students are apathetic or traumatized by math?

I plan to come back to these topics and more, so stay tuned!

Two Summer 2011 Middle School Math Programs

2010-12-29T23:05:00.000-05:00

I just heard from Steve Maurer, the academic director of MathPath, a summer math program for middle school students. His program has run since 2002, and the math instructors are something of a dream team, featuring some famous mathematical names such as John Horton Conway (well-known genius mathematician), Titu Andreescu (who runs the American Math Competitions) and Sam Vandervelde (author of Circle in a Box).

This program seems ideal for middle school students who are already budding mathematicians. They've previously recruited students who have done well at math competitions, but are now making a concerted effort to reach out to math circle participants who are motivated by the excitement of mathematics itself. I hope this new effort succeeds! Teachers and math circle organizers: please send some great students towards MathPath!

MathPath

URL: http://www.mathpath.org/

Dates: Sunday, June 26 to Sunday, July 24, 2011

Location: Colorado College, Colorado Springs, CO

Another really interesting summer math program for middle school students that I'm in contact with is brand new this year, the Summer Program in Mathematical Problem Solving (SPMPS). This program also has ties to the high school level Canada USA Mathcamp (http://www.mathcamp.org/) but its target audience is quite different: underserved New York City middle school students with talent in math. One goal is to set these students on their own math path and to open up access for them to the wealth of high school math programs enjoyed by students at more privileged schools. I wonder how many mathematicians will eventually come out of SPMPS?

I've been helping the director of SPMPS, Dan Zaharopol, to network with underserved middle schools in NYC, and he's off to an exciting start. Dan has sponsorship from the Art of Problem Solving Foundation to start his program at Bard College this summer. It promises to be a great program, and I wish him luck!

Summer Program in Mathematical Problem Solving
UR: http://www.artofproblemsolving.org/spmps/
Dates: Monday, July 11 to Monday, August 1
Location: Bard College, Annandale-on-Hudson, NY

Nice LaTeX Reference

2010-10-19T11:06:00.000-04:00

I just noticed that Wikibooks features a decent LaTeX reference: http://en.wikibooks.org/wiki/LaTeX

The Grasshopper Problem

2010-09-20T16:51:00.000-04:00

This is my newest problem, to appear in the October 2010 issue of La Voz.

What is this picture asking? Ben suggested that instead of bold black squares, I should have indicated the squares with something especially significant to grasshoppers. Something tasty or something dangerous?

Where did this come from? This problem is related to a derivation of the formula for Primitive Pythagorean Triples: Find all relatively prime natural number triples (a,b,c) that satisfy the equation

The first step in my favorite elementary number theory proof is to show that not both a and b are odd. This would imply that c is one of the shaded squares above: 2, 6, 10, 14, ...

Does the grasshopper land on a (boxed) square?

Bard Math Circle meets in the Kingston Library

2010-09-04T12:45:00.000-04:00

I'm helping the Bard Math Circle start up math circle activities in the Kingston Library.

Each session will include time for puzzles and games, creative mathematical problem solving, and a hands-on math activity with a take-away.

Meetings this fall: 1pm to 3pm on each second Saturday (September 11, October 9, November 13 and December 11) at the Kingston Library (55 Franklin Street, Kingston NY, 12401).

First meeting: 1pm to 3pm on Saturday, September 11, 2010.

Please forward these posters (or download and print).

Poster (English) Bard Math Circle in Kingston

Poster (Español) Circulo de Matemáticas en Kingston

This math circle activity is also mentioned in the August 30, 2010 Kingston Community Builder.

I'm grateful to Margie Menard, director of the Kingston Library, for sending out this press release.

Thanks for helping me get the word out!

Following Math Blogs

2010-08-26T18:37:00.000-04:00

I've slowly been entering the world of math blogging, first by reading blogs, and now by posting a note occasionally on this blog. There is an amazing world of math bloggers out there, some I'm proud to know in person, and so many worth reading.

Here's a post claiming to be the top 25 classroom math bloggers: http://www.onlinedegrees.org/top-25-blogs-for-math-in-the-classroom/, which may be the best list to get started. Give it a read!

Rubik's Cube Conversation over Breakfast

2010-08-11T09:07:00.000-04:00

If you read my previous post then you know that I'm at HCSSiM for the second half of the second half, teaching a mini on the Fundamental Theorem of Algebra. I'm very impressed with everything, from the teaching faculty to the students, and am really enjoying soaking in interesting mathematics every day. There is really high level mathematics going on here, and the students are truly immersed in mathematical knowledge and culture.

Conversation at breakfast with Lucas and Gabe.

I walked into a conversation about Rubik's Cube records this morning at breakfast. Talking about new cube records, Lucas complained about kids who ask him if he solves the cube by just doing the same sequence of moves over and over again. Of course not! This would only work in a cyclic group!

However, I argued that it is possible, if you look at it another way. Suppose that g is a long sequence of moves that traverses through all possible cube positions. Then you only have to do the sequence g once, and somewhere along the way, you'll have solved the cube. Notice that the end result of performing the moves in sequence g is the identity permutation on the cube.

We can improve this by finding a cube permutation g that generates a large cyclic subgroup of the cube group. Let G be the cyclic subgroup generated by g. If we can express g as a long sequence of cube moves that traverses through a complete set of coset representatives of G, then we have the cube neophyte's dream: a sequence of cube moves, that it you do over and over again, will eventually solve the cube (of course, in the worst case scenario you'll move through all possible configurations of the cube, but I'm not making any claim about the efficiency of this method!)

We finished our breakfast conversation by posing a more reasonable problem: try this for a small group.

1. Show that g = (1, 2, 3)(4, 5) is an element of S_5 of maximal order.
2. Find a sequence of 120/6 = 20 permutations s_1, s_2, ..., s_20 whose product is g, and whose partial products (s_1), (s_1 s_2), (s_1 s_2 s_3), ..., (s_1 s_2 s_3 ... s_20) is a set of coset representatives of the cyclic subgroup .
3. Solve problem 2, where each s_i is a transposition.
4. Solve problem 2, where each s_i is a transposition of adjacent elements.

This seems like a good start to investigating the breakfast conversation problem. Let me know what you think!

HCSSiM 2010: Alive and Well!

2010-07-18T01:00:00.000-04:00

I visited the Hampshire College Summer Studies in Mathematics (HCSSIM) program for the annual YP Day, and am happy to report the program is as vibrant as ever this summer.

Yellow Pigs Day

For those not familiar with HCSSiM lore and what YP day is, the initials YP stand for Yellow Pig, one of the two mascots of the program. David Kelly, the director of the program, has perhaps the largest collection of yellow pigs in the world.

The other mascot of the program is 17, the favorite number of scads of summer studies students and alumni. YP Day (Yellow Pig's Day) falls each year on July 17th, and is the traditional time for alumni to revisit the campus and reconnect with HCSSiM.

This Year's Faculty

Arriving at HCSSiM on Friday in time for lunch, I met the faculty. In addition to Kelly, we have Rob Hochberg and Josh Greene also leading the workshops. Rob, a computer scientist, and I go way back to the 1980's when we were both summer studies students, but this was my first time meeting Josh, a Columbia University mathematician, who attended HCSSiM in the 1990's. To round out the crew we have the amazing:

Gabe, recent University of Chicago graduate
Misha, math graduate student at CUNY
Nadine, math graduate student at the University of Pittsburgh
Achyut, Hampshire College physics major
Peter, Harvey Mudd math major
Nate, U Mass math major
Julia, recent Hampshire College graduate
Emil, Harvey Mudd math major

and of course the indefatigable Susan Goff, who has been the program assistant for over a decade.

Prime Time Theorem

I got my chance to meet the students later that afternoon, when I explained How to Solve a Cubic to them as Friday's Prime Time Theorem speaker. The students are as sharp as ever, and an enthusiastic audience. I ended my talk with some open-ended questions.

Is there a quartic formula?
Is there a quintic formula?
Does every polynomial have a zero?

Although most of the summer studies students know the general answer, I hope that these questions and my talk will pique their interest for my Mini later this summer on the Fundamental Theorem of Algebra. Apparently another Mini will focus on Galois Theory, so HCSSiM definitely has this topic covered!

After dinner I had a chance to talk with the students during their problem session. They're enjoying some Algebraic Number Theory, Cardinality Proofs, and Group Theory.

Saturday, July 17th

Yellow Pig's Day fell on a Saturday this year, just perfect timing for the traditional Ultimate Frisbee Game, Kelly's talk on The Mathematical and Social History of 17, the Yellow Pig Cake and the Singing of Traditional Yellow Pigs Day Carols.

Saturdays start off with the morning workshops, though. I was able to wander between the three workshops and observe a wonderful proof of Wilson's Theorem, a listing of Group Theory axioms (after two weeks of examples), and some interesting Cardinality Proofs.

Mathcamp

Surprisingly close this summer is Mathcamp, another wonderfully rich math environment for high school students. Math camp is just a 15 minute drive from Hampshire College at Mount Holyoke College, I visited Dan Zaharopol's class on Real Analysis, and Alison Miller's class on p-adic numbers.

Among the more than 100 Mathcamp students, there are several who previously studied at HCSSiM, and I was intrigued discussing their positive experiences at both programs.

Ultimate Frisbee Game, HCSSiM 2010 vs. HCSSiM <>

My visit to HCSSiM 2010 ended with a grueling game of ultimate in excruciating heat. The alums were outnumbered three-to-one, but did well until the 2010 students developed a strategy that took advantage of their strength in numbers: they lined up and passed the frisbee down the field. Luckily, the aging alums still managed to come out ahead by dividing each team's score by the number of players, and then multiplying by the average age.I drove back home soon after dinner, and enjoyed a beautiful double rainbow all the way through the Berkshires. I'm looking forward to returning in a few weeks to lead a Mini on The Fundamental Theorem of Algebra.

¡Juguemos Kenken!

2010-04-30T12:09:00.000-04:00

(This column originally appeared in the July 2009 issue of La Voz, in Spanish.)

Welcome back to the puzzle column. This time we have two new kenken puzzles for you. The first one is a 4x4 kenken that uses only addition, and the second a 5x5 kenken made up entirely of multiplication.

For your convenience, we repeat the rules of kenken:

Similar to Sudoku, your task is to fill in the 4x4 grid using only the numbers 1, 2, 3 and 4 (and the 5x5 grid using only the numbers 1,2,3,5 and 5) in such a way that:
1. Each number appears only once per row.
2. Each number appears only once per column.
3. The numbers in each cage, when combined with the operation given, yield the target number.

Since kenken means “wisdom squared” in Japanese, we know you'll get a little smarter by successfully completing these puzzles!

Multiplying

The second kenken puzzle is entirely multiplication, which means that a good strategy to use is factorization. Look at the top middle cage, for example. We have to find three numbers whose product is 20. Multiplication has the amazing property that the order of the factors doesn't matter, so listing the factors in non-decreasing order, there are four ways: 20 = 1×1×20 = 1×2×10 = 1×4×5 = 2×2×5. Of course we immediately eliminate the first two ways, since we are only allowed to fill in the grid with the numbers 1, 2, 3, 4 and 5.

Using factorization on the bottom right cage is even more effective: 25 = 1×1×25 = 1×5×5. The first factorization uses 25, which is too big. The second factorization has a repeated factor, which in combination with rules 1 and 2 above, allow us to conclude the unique way to fill in the cage.

Factorization leads to some amazing patterns. Consider the cage in the middle of the puzzle that prompts us to express 12 as the product of two numbers. The solutions are 12 = 1×12 = 2×6 = 3×4. The numbers that appear, 1, 2, 3, 4, 6 and 12, are called the divisors of 12. There is an amazing way to arrange the divisors of 12, shown below. Moreover, by counting 1 and the number 12 itself, 12 has 6 divisors.

This figure is called the divisor lattice of 12. Notice that moving up and to the right (northeast) is equivalent to multiplying by 3. Likewise, moving up and to the left is equivalent to multiplying by 2. From a tilted perspective, the divisors of 12 form a 2×3 grid.

Problem C: Draw the divisor lattices for the numbers from 1 to 20. What do 2, 3, 5 and 7 have in common? Can you draw the divisor lattice for 30? Is 1 a prime number? What other patterns do you notice?

Problem D: What is the smallest number with exactly 2009 divisors? (count 1 and the number as divisors).

If you enjoyed these puzzles, I'd love to hear from you. Please send in your solutions to any of the problems A, B, C or D as a comment.

¡Juguemos Kenken!

2010-02-28T12:12:00.000-05:00

(This column originally appeared in the June 2009 issue of La Voz)

KenKen es un puzzle nuevo inventado por Tetsuya Miyamoto, un maestro de matemáticas del Japón. KenKen significa “sabiduría al cuadrado” y se lo presenta como una variante del Sudoku. Este mes les traigo dos juegos de KenKen de 4x4. Uno más fácil con sumas solamente, y otro con más operaciones. ¡Y unas preguntas extra para los amantes de los rompecabezas matemáticos!

¿Qué es un juego KenKen?

Cada juego KenKen de 4x4 consiste en una grilla cuadrada de 4x4 dividida en grupos llamados jaulas o regiones. Las regiones vienen en diferentes tamaños: 1, 2, 3 y a veces 4 cuadrados. Cada región indica el número al que se debe llegar y una operación.

Por ejemplo, acá hay una región hecha con dos cuadrados, con un “7+” chiquito escrito en un rincón. Esto quiere decir que el número al que se debe llega es el 7 y la operación es la suma.

La tarea es llenar la grilla de 4x4 usando solamente los números 1, 2, 3 y 4, de tal manera que:

Cada número aparezca sólo una vez por fila.

Cada número aparezca sólo una vez por columna.

Los números en cada región, al combinarse con la operación dada, den por resultado el número al que se debe llegar.

Volviendo a nuestro ejemplo, vemos que hay exactamente dos maneras posibles de completar la región: 7 = 3 + 4 y 7 = 4 + 3 son las únicas posibilidades. La elección correcta dependerá de las primeras dos reglas, y las posibilidades para las otras regiones.

A veces una región tiene un solo cuadrado. En ese caso, el número final es dado sin ninguna operación. Esto es un regalo gratis, y es una buena manera de comenzar a completar el rompecabezas.

Esto es todo lo que necesitas para resolverlo, además de imprimir esta página para poder escribir, claro. ¡Así que a jugar KenKen!

Problema A

Problema B

El cuadrado latino

Un cuadrado de 4x4 completado con los números del 1 al 4 se llama cuadrado latino si satisface las dos condiciones de arriba. O sea, un cuadrado es latino si usa cada número sólo una vez en cada columna y en cada fila.

Problema C: ¿Cuántos diferentes cuadrados latinos puede haber si la primer fila es 4 2 1 3?

Puedes construir tu propio juego de KenKen empezando por el final. Primero, completa una grilla de 4x4 con un cuadrado latino. Segundo, dibuja regiones dentro de la grilla y elije una operación para cada región. Puedes empezar con la suma, para hacerlo fácil, pero los juegos de KenKen también tienen resta, multiplicación y división.

En cada región, combina los números con la operación que elegiste para determinar el número al que hay que llegar.

Ahora que tienes la solución, borra los números del cuadrado latino original, y deja los resultados y la operación de cada región. Si tu puzzle tiene una solución única, ¡tienes un juego de KenKen!

Problema D: Envíanos tu mejor puzzle de KenKen original

Si te gustó jugar al Kenken, envíanos tus soluciones a alguno de los problemas, A, B, C o D. Agradeceremos tu participación en futuros Blogs.