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Need help with a math extended essay topic

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I'm having trouble coming up with a good topic that feels like I could write my extended essay on. I want to do something in the field of geometry, or at least in logical problem solving, but I'm finding it difficult to pull any ideas to expand on.

 

One idea I did find that I was interesting in is the Heilbronn Triangle Problem. It's an interesting problem, but it's not actually solved, nor is there a lot of information on it, and I don't even know how I could expand upon it in a way someone hasn't already done (different shapes and more dimensions have already been taken). I just feel stuck with that, and I can't find any other topics that I'm particularly interested in.

 

So is there some way I could make a paper out of this, or are there any similar topics that would be good to use?

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I haven't looked at the problem too closely, but in general for these maximization problems you can try to find upper and lower bounds. Essentially you start with some very loose bounds, then slowly improve them and get closer to the actual result. Alongside that, you may work with fixed, low values of n to begin with to get an idea of strategies that may generalize for higher values of n - or even just stick with fixed values and try to find optimal configurations (possibly even mention/use numerical methods to algorithmically find the best configuration, say). Another option is to take a simplified version of the problem (special cases) and work from there.

 

Even if tighter bounds and results are already known, you could probably have your own investigation and it'll be fine in terms of originality - though I'd recommend asking your supervisor about this. The problem does look pretty difficult though, so it depends on how much progress you can make. It seems like the circle case is the easier one given the symmetry, so that may be a good place to start.

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