# Rambling Thoughts

## Random thoughts from 2012 ARML

Last weekend was 2012 ARML. It was my fifth ARML, and third as coach of the SFBA teams. I remember my first trip to ARML, with just a single team. This year, ARML has become a 7-team, 4-day expedition. This was also one of my most entertaining trips to ARML. Watching SFBA troll the Friday evening talent show was amazing.

As for results, SFBA’s teams did amazingly well: A1 and A2 were 3rd and 7th nationally, the first time that any organization has had two teams in the top 7 at ARML. Next year, we’re aiming for 1st and 2nd on the national scoreboard.

Some random thoughts from ARML:

• Tie-dye was an amazing shirt color. Pink was pretty cool too.
• I should try to avoid losing my bed next time.
• Flip, Flop, Fliegerspiel
• Bif! Bif! Bif!
• I am Chen!
• Yee!

## ARML

Sorry about not posting for the past few months. I’ve been very busy recently. With the start of the summer, I should be able to post much more frequently.

I have been a coach for the San Francisco Bay Area ARML team for the past two years. Every year, people ask me why I help with ARML, and the answer is always the same: because it’s fun.

I enjoyed my first trip to ARML in 2008, and ARML has gotten better every year. Unlike most contests, ARML is heavily team-based, and as such, it is a great way to meet interesting people. ARML tends to build a sense of community that is lacking from most math events. Over the past few years, I’ve met many students and coaches at ARML, and they are a major reason why I go back to ARML every year.

Another big part of coaching ARML are the practices that we run every spring. It’s always great to see the gathering of people with similar mathematical interests. I believe in teaching math as a way to learn and understand something more thoroughly, and I hope to do more teaching at ARML practices in future years. Doing math is often most enjoyable as a social activity, and I think that our practices fully demonstrate that feature.

Here’s a list of interesting things that happened at ARML, in no particular order:

• Green sharpies
• Easy button thieves
• A disappearing sword
• Brilliant power round solutions
• Paper balls and paper airplanes
• Math!

## ARML

ARML (American Regions Mathematics League) is an annual math contest that takes place every year at four sites around the country. It’s one of my favorite math contests.

ARML is different from most other math contests because it requires teams to travel to one of four central sites. This means that teams have to travel (some times for very long distances) to participate. Inevitably, this means that participation at ARML is not especially high; only 124 teams of fifteen people (in 2010), or less than one percent of the more than 200000 people who participated in the AMC. Teams that do participate at ARML, however, are much more organized; most teams organize several practices before the contest, while some teams prepare for ARML year-round. The team aspect of the contest (3 out of 4 rounds have a team component, together constituting 50% of the total team score) also makes ARML stand out. Indeed, ARML is primarily a team contest, and the team results every year are watched much more closely than the individual high scorers. As a result, ARML is much more social than most other math contests, giving participants a chance to work closely with their team and to meet people from other teams. This social aspect of ARML is what makes ARML enjoyable.

Mathematically, ARML feels different from most other contests. The ARML style is to have problems that look hard but are made much easier through clever tricks. Often, these problems can be solved without being clever, but cleverness can often yield very simple solutions. As an example, consider this year’s coffee mug problem: [EDIT (17 September 2010): Updated with the original wording.]

Let $P(x) = x^2 + 2010 x + 2010$, and let $r$ and $s$ be the roots of $P$. If $Q$ is a quadratic polynomial with leading coefficient 1 and roots $r + 1$ and $s + 1$, compute the sum of the coefficients of $Q(x)$.

This can be bashed out without the quadratic formula or with Vieta’s formulas, but the clever solution is much simpler:

Because $Q$ is monic,
$Q(x) = (x - (r+1))(x - (s+1))$
$= ((x - 1) - r)((x - 1) - s) = P(x-1)$. The sum of the coefficients of $Q(x)$ is $Q(1) = P(0) = 2010$.

Together with the clever tricks, ARML imposes a short time limit on its tests, making it difficult for contestants to finish the team and individual tests. The relay round’s three-minute and six-minute time limits provides a further emphasis on speed. As a result, ARML problems (with the exception of the proof-based Power Question) are usually not very hard mathematically; they’re just hard to do under severe time pressure. This short time limit, along with the small number of questions on ARML, make the results of the contest subject to a lot of random noise. The top teams are usually within a few points (out of 300 points total) of each other, so a five-point team round problem could easily change a team’s ranking by several places. The individual round has only ten problems, so achieving a national high score (usually at least eight problems) requires making no computational errors. Thus, the list of individual high scorers is somewhat meaningless; high scorers often do not repeat as high scorers when they return, and high individual scorers are often as much a result of luck as of mathematical ability.

Evaluating ARML based on the criteria that I listed in my post on Contest Math, we obtain mixed results. ARML, perhaps more than any other contest, highlights the social side of mathematics and hooks people into liking and doing math; though it’s hard to start doing ARML, it’s even harder to stop. Mathematically, though ARML does reinforce the importance of creativity, it does not give particularly challenging problems, and ARML results do not really constitute a good measure of mathematical achievement.

I enjoy ARML primarily based on the social part of the contest, and many people agree with me; the highlight of ARML for some people is the long bus ride to the contest. Indeed, I think of ARML primarily as a social event, with some mathematics to make it look like the participants are actually doing something important.

## Contest Math

Many people first become interested in mathematics through contest math in middle school or high school. Others, however, progress to higher level mathematics without participating in the contests at all. What, then, is the role of math competitions in attracting people to mathematics? Is contest math beneficial for doing math beyond competitions?

This, of course, depends on what we mean by “contest math”. Of course, a competition to produce research results would certainly be different from a mental arithmetic contest. For now, I define contest math to be in between these two extremes; the problems are not straightforward, but they are not at the difficulty of research questions. In particular, I classify Mathcounts, ARML, AMC, USAMO, and Putnam in the category of contest math.

One positive effect of contest math is that it tricks students into liking and doing math at an early age. The mathematics education system in America is not conducive to positive impressions of mathematics, and contest math provides an alternative world where problem solving ability is emphasized above rote computation. For example, the Mathcounts contest hooks many students in middle school, providing an introduction to math and leading them to a continued interest in mathematics.

Contest math hooks students partially through its competitive nature. Unlike the typical mathematics classroom, contest math provides a challenge. There are always harder problems and puzzles to consider, and students never run out of room to explore. The feeling of solving a difficult problem is always exhilarating, and pushes the student to do more problems. Furthermore, the contest math community is full of students who are always competing to be the best. The rankings at each math contest push students to work harder than they would if left alone, and the competitive spirit injects energy and excitement into mathematics.

Along with competition, contest math brings together a group of people who are interested in the same thing: Solving math problems. This gives contest math a social aspect as well, challenging the conventional notion that doing math and being social are mutually exclusive activities. A good example of this phenomenon is the ARML contest. In many regions of the country, ARML brings together the best students and gives them an opportunity to get to know each other through practices and an annual trip to the national competition. This social aspect of contest math even extends to the Art of Problem Solving forum, for example.

After contest math hooks its victims, it also indoctrinates them in good mathematical principles. In particular, since contest math values ingenuity over knowledge, it ends up very a good teacher of problem solving. This is important because the ability to think about and solve problems is the key to success in higher level mathematics. The best mathematicians are the ones who can make connections and devise clever arguments. In this way, through the bait of math competitions, contest math pulls students toward real mathematics.

Despite the positive effect of producing more mathematically-minded humans, contest math has many deficiencies. In particular, contest math is not representative of how mathematics is actually done. In reality, mathematical research requires much more knowledge than Euclidean geometry, and pure ingenuity often isn’t sufficient. Also, math competitions teach students to expect solutions to their problems within minutes or hours. Though this is much better than the seconds-long attention spans of many high school math students, it is far from the days, months, or years that are needed to solve difficult research problems in mathematics. Thus, in some sense, contest math forms a different (but related) subject from mathematics, and by doing contest math exclusively, students sometimes lose sight of what mathematics is actually about.

Thus, contest math taken to excess is almost never a good thing. By overtraining themselves for math competitions, students do nothing to improve their mathematical ability. Some students even become resistant to the idea of doing any math beyond the contests that they’ve focused on for years, rendering their mathematical talent useless. This is especially true for more computationally oriented contests such as Mathcounts. Though they provide good introductions to math, they are too far from actual mathematics to be meaningful for any more than a year or two. In this way, contest math can often be harmful for the future mathematician.

In the end, we see that contest mathematics is good for providing an introduction to the world of mathematics to those who haven’t seen it before. However, it cannot replace mathematics, and eventually, math competitions must lead into higher level mathematics. Even among math contests, some are more meaningful than others; not all math contests are equal. But that’s a topic for a future post.