Mathematics Creative Problem Solving Competition

[Orestes Hardy]

TheInstitute of Creative Problem Solving for Gifted and Talented Students'mission is to enhance the problem solving skills of exceptionally talented students using advanced mathematics and creative thinking. The Institute strengthens our nation's development of young leaders and role models in mathematics, science and technology to allow them to contribute greatly to society. The Institute is tuition free, providing equal access to all qualified students from Long Island and promoting underrepresented populations in the mathematics, science and engineering fields.

I've been trying to figure out the divide between "good" and "great" mathematicians, and one metric I see repeatedly is "mathematical creativity". Great mathematicians seem to be able to pull new constructs out of thin air, whereas a merely "knowledgeable" mathematician can only mull about in the immediate facts.

This is where I feel I'm at personally in my mathematical career: mulling about in trivialities. I can derive maybe a few trivial implications from the assumptions, but often I need help in solving a problem, and once I see the solution's ingenuity I get the nagging feeling that I never would've been able to come up with those ideas myself.

I'm learning a lot of "things", but just learning the things hasn't seemed to improve my problem solving ability. Even solving math assignments, contest problems, working through proofs again and again, memorizing lemmas using spaced repetition software, all of it seems to have only made a marginal improvement. No matter how much math I seem to cram into my head, my problem solving creativity seems to remain at just about the same level.

Mathematical creativity is tricky, it's not like traditional creativity: it's paradoxically "creativity with constraint". How can one simultaneously think "outside the box", but "within the lattice" of mathematically reasonable ideas?

There is likely no formula for creativity, otherwise it would be common knowledge by now. Some people will just be genetically more insightful than others. I think it would be unwise to end the discussion there, though.

Not everyone may be genetically built for bodybuilding, not everyone may be able to achieve the Schwarzenegger physique, but this doesn't invalidate the breadth of scientific research on bodybuilding. We've extensively developed bodybuilding techniques, we have rigorous means of discussing it.

I see the mind in the same way, with creativity as the muscle. Who's to say we there isn't a means of improving creativity like there's a means for building muscle, or a means for increasing our knowledge? Why shouldn't we expect that we simply lack the tools to discuss mathematical creativity?

It partially depends on what kind of "creative" you want to be. Gian-Carlo Rota divided mathematicians into two groups, "Problem-Solvers" and "Theorizers". The problem-solvers solve new problems and the theorizers rethink old, already-solved problems until they are so obvious that they don't really need solutions.

I think one thing that stymies creativity in mathematics is the emphasis on proofs. I think that proofs are one of the things that makes mathematics great, but the hyper-focus on them actually distracts from teaching others to think great thoughts. You almost never read about how someone first started thinking about something, or how they imagined it that allowed them to write the proof in the first place. These things that take place in the imagination are rarely discussed, and, instead, mathematics focuses on the end-result, the proof, rather than the process that got there.

I think the key to creativity of both types (problem-solving and theorizing) is a willingness to think about a problem in a new way, or to take a new perspective on a field. It's great to know all there is to know about a field, but sometimes this becomes a trap, such that you get used to thinking about a field in the same way as everyone else. You get sucked down that hole, and don't even realize you are there. For this reason, in Meta Math, Chaitin suggests that newcomers are sometimes the best people to advance a field. They aren't stuck in the same way of thinking as everyone else.

But, I think that there is another way out of the hole - being widely read. And by this I don't just mean mathematics. Read philosophy, physics, theology, engineering, poetry. Many mathematical tools were developed by the physicists long before they were provable by the mathematicians, and they are much more likely to relay the reasoning that led to their ideas. These things can give you a new perspective - an ability to see things that others don't see.

One other method I suggest for creativity is to not worry about the impossible. There are, indeed impossible things. However, most things that are "impossible" are only impossible due to certain assumptions (which are reasonable to everyone at the time, and sometimes so obvious that they are completely unstated). One can be creative by imagining a world without some combination of those assumptions, and seeing what the new world looks like. Even if removing the assumption is impossible, the practice of imagining the world like this is helpful. It helps you practice in seeing through to what alternate sets of assumptions might look like, and helps your brain to recognize situations in which changing assumptions makes the world look more correct.

If you are a problem-solver, the practice of removing assumptions is extremely helpful. If you are a theorizer, I would suggest looking into computer programming. In computer programming, there is a concept called "refactoring", where you re-analyze existing code to see where there are redundancies and things can be reworked in a way that is simultaneously simpler and more powerful. These tools and techniques can often be applied to mathematics to rework existing mathematics into a form that is likewise simultaneously simpler and more powerful. See here for an example.

Carnegie Mellon University Mathematical Science Professor Po-Shen Loh has been using math competitions to teach creative problem-solving to students for most of his career. Now, he's advising a high-stakes competition for artificial intelligence models.

XTX Markets launched the Artificial Intelligence Mathematical Olympiad Prize (AIMO) in November 2023. The $10 million challenge fund is designed to spur the creation of a publicly shared AI model capable of winning a gold medal in the International Mathematical Olympiad (IMO), the premier math competition for high school students worldwide.

"In over 100 talks, I have interactively queried ChatGPT on math topics that are freely proposed by broad audiences," Loh said, referring to a system developed by OpenAI that allows users to generate content, answer questions and provide detailed explanations.

"Week after week, I have observed that even the current version of ChatGPT usually responds to math discourse at the level of a strong high school student, across a wide variety of topics. But it also sometimes stubbornly makes mistakes, and then sticks with those mistakes in ways that people would not," he said.

In February, XTX announced that Loh and other prominent mathematicians along with AI and machine learning specialists will serve as an advisory committee to support the development of the AIMO Prize including advising on appropriate protocols and technical aspects and designing the various competitions and prizes.

Along with Loh, the group includes Timothy Gowers and Terence Tao, both winners of the Fields Medal, one of the most prestigious awards in mathematics; Dan Roberts, an AI researcher at Sequoia Capital and MIT and a published expert in machine learning; and Geoff Smith, the former president of the IMO.

"We are delighted to welcome such esteemed members of the mathematical community to the advisory committee of the AIMO Prize," said Eoghan Flanagan, Director of the AIMO Prize. "I look forward to working with them on developing and growing the prize."

An IMO silver medalist during high school, Loh served as the national lead coach for the U.S. team for a decade. During that time, the U.S. placed first four times. In 2019, he received the U.S. Presidential Early Career Award for Scientists and Engineers, and in 2002, he received a Hertz Fellowship.

"International Math Olympiad problems are difficult because they involve many sequential layers of reasoning," Loh said. When teaching, Loh focuses on the power of mathematics, and how it provides a framework for thinking so that students focus less on following steps and more on problem-solving. In the same way that students can work collaboratively through trial and error to answer problems, AI models can do the same. "The next big breakthrough will come from solidifying the concept of proof, and this prize supplies a useful target for that research."

The first publicly shared AI model to enter an AIMO approved competition and perform at a standard equivalent to a gold medal in the IMO will receive a $5 million grand prize. There also will be a series of progress prizes, totaling up to $5 million for publicly shared AI models that achieve key milestones.

Competitions have become opportunities for researchers to shape the next frontier of AI research and advance responsible AI-driven solutions to critical societal challenges. For example, in recent years Carnegie Mellon researchers have participated in XPRIZE competitions that have incentivized the use of AI for breakthroughs in digital learning, landing and operating a robot on the moon and Defense Advanced Research Projects Agency (DARPA) Challenges intended to accelerate the use of autonomous vehicles in challenging environments. A team from Carnegie Mellon's Department of Chemistry and the University of British Columbia recently won a prize to advance Parkinson's research through using computational methods.

Gowers, a professor of combinatorics at the Collge de France, said that while machine learning methods have had spectacular success in several domains, their performance on tasks that involve multi-step reasoning still lags behind.

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