Possible research topics for high school I'm a junior in high school, and this year I have the opportunity to work on a STEM research project which I will enter in my school fair in February 2018. I know pretty much for certain that I want to do something with pure math, but there are so many sub categories, I am lost on what I want to research. So far, I have learned algebra 2, high school level geometry, and calculus (the AP BC course material). What would be some interesting topics that would be suitable for me to do a research project on?
 A: We can't answer the 'interesting' criterion for you, but that is the most important. If you find something interesting but difficult, it won't seem so difficult because you'll be getting a kick out of it. But I can suggest some areas where the bridge from everyday to math is easy, and that's probably a good place to start. For example:


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*symmetry in nature. Flowers, organisms etc. Leads to some nice number theory and things like Fibonacci. If you like art, you can extend to things like the golden ratio, the art of MC Escher and so on. 


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*math in music. Look at scales and intervals and both western and non western musical systems. Nice tie in to physics of waves of  course and a great excuse to listen to all different yes of music... 


*number puzzles. This is a very fertile field. If you  an program a computer, can you write a sudoku solver? What strategies work? Why? More classically, simple statements about numbers give rise to some really deep number theory and math. Goldbachs conjecture, Fermat etc. 


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*crypto. Very relevant to the modern world. What makes a good system? Why are some systems considered better than others? 

*graph theory. Did you know that Facebook and other social media sites are built on graph theory? What mathematical properties of the whole 'friend network' exist? How does graph theory extend to the 'cool kids' network at your school? 

*math and society. Why is math considered hard? Is it cultural? If you are lucky enough to attend a diverse school, what attitudes to math exist in different groups? You might be surprised. 



If you have time, I thoroughly recommend getting anything by Martin Gardner or Douglas Hofstadter, the former as a very fertile source of math games that spur investigation and the latter as a way of thinking mathematically about the world. His book Godel Escher Bach is legendary. 
Edit: concerning how to find something that no one has done before. For a high school / undergrad project, chances are that the basics will have been covered, but the chances that your particular application of the project has been covered are much lower (and therefore better for you). For example: the math behind the golden ratio is well explored. The math behind the golden ratio in architecture too. The math behind the golden ratio in the  architecture of your town or state? Much less so.
A: A really nice problem here:
Describe the set of real numbers such if $x$ belongs to the set, then $x^{-1}$ also belongs to it, and both $x$ and $x^{-1}$  share the same decimals. Hint, you know that 1, $\phi$ and $\phi^{-1}$ they belong to the set (being $\phi$ the golden ratio). Are they the only three numbers with that property, or are there more? (Check right now the decimals of $\phi$ and its inverse!)
Rules: do not watch here in Math.SE. Try by yourself.
Hint plus motivation: this is a really wonderful problem because even when it is apparently hard, you already have all the required knowledge. Just some calculus and think. And when the solution hits your mind... You understand how easy was, and all the implications it has. Oh! Start just with the positive real numbers and forget about the negative ones, everything will be clear at the end.
Raising up this to the category of a work: after you've solved the problem, look for more information. You'll understand the connection with the golden ratio and geometry.
Good luck!
