What Math Problem Should I Tackle for My Summer Research Project?

[Mr.Gronka]

Hi everybody,
I am currently at a summer program hosted by my home state of Georgia (USA) called the Governors Honor's Program. Right now we are preparing for our final project that's supposed to take up the remainder of our time here (roughly three weeks). My partner and I are supposed to select a math problem that sort of "hasn't been solved yet" and, well, solve it. Some students do very well with their choices, and the teachers keep telling us about this group last year that was published on MathWorld.com. I'm not looking for any fame, but I would like some suggestions.

Some of my friends have chosen topics such as programming a simulator of sorts, modeling a spherical triangle, or determining if the Euler-Mascheroni constant is irrational. If we don't solve our problem, we are supposed to report what we learned.

edit: The problem can be physics-based, and our topic is due tomorrow at 9;50, although we're in no big rush. I've had up through AP Calculus, but the topic doesn't have to include calc.

Thanks,
Taylor

 

[mighty2000]

Hi Taylor,

It sounds like you have a great opportunity to explore and solve a math problem at your summer program. Here are a few suggestions that might interest you:

1. The Collatz Conjecture: This is a problem in number theory that has been around for over 80 years and still remains unsolved. It involves a simple iterative process and has been studied by mathematicians and computer scientists alike. You could try to explore different approaches to solving this problem and see if you can come up with any new insights.

2. Goldbach's Conjecture: Another famous problem in number theory, this conjecture states that every even number greater than 2 can be expressed as the sum of two prime numbers. Despite being tested for numbers up to 4 x 10^18, it has not been proven to be true for all numbers. This could be an interesting problem to investigate and see if you can find any patterns or exceptions.

3. The Traveling Salesman Problem: This is a classic problem in computer science and operations research, and it involves finding the shortest possible route that visits a given set of cities and returns to the starting city. It has applications in various industries, such as logistics and transportation, and has been studied extensively. You could try to come up with a new algorithm or approach to solving this problem.

4. The Four Color Theorem: This is a problem in graph theory that states that any map can be colored with only four colors in such a way that no two adjacent regions have the same color. It was first proposed in 1852 and was finally proven in 1976. However, there are still many variations and extensions of this problem that have not been fully explored.

5. The Monty Hall Problem: This is a famous probability puzzle that has been featured in various TV game shows and has sparked much debate and discussion among mathematicians and statisticians. It involves a scenario where a contestant is given the option to switch their choice after being shown a wrong answer. You could try to analyze different strategies and see if you can come up with a solution that maximizes the contestant's chances of winning.

I hope these suggestions help and good luck with your final project!