What's fun in math

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Hi,
I want to find something in mathematics that's fun to work on but also try to solve something that is still a mystery and unsolved to people. I enjoy prime numbers, geometry, algebra and trigonometry but there's no mystery. It's all been worked out. What I'm after is something that's interesting and hasn't been solved yet. Does anyone know of such a thing? All ideas welcome!

Unsolved mysteries in math are few and far between--and the common denominator seems to be that the people who contribute with new discoveries, are young (below 25), hard-working, and do not see the limitations other people see. Furthermore, their discoveries are typically based on the discoveries of other people; Leonhard Euler didn't discover the natural logarithm, he just finished where others left of.

There are a few problems you could take a look at, though:

http://en.wikipedia.org/wiki/List_of_un ... athematics

Just being honest.

May I recommend Graph Theory. It is an interesting field and it is often the first port of call for research. Have a look at this to get started: http://diestel-graph-theory.com

There are almost no perquisites also so thats a bonus.

The basic areas are worked out, and the few open problems are problems that have proven very difficult. However, as problems are solved, new problems and areas are discovered. If you go into the details, and the details of the details, there are countless open problems. Hundreds of thousands of mathematical articles are written per year, most containing new, even if small, results.

I second graph theory, and combinatorics in general, as an area that requires relatively little prerequisites - in contrast to fields such as algebra or topology where you need years of study to even understand new research areas.

Video games.

Big time math in them, matrix manipulation. Wish I had the time to learn it. Its partially about multiplying matrix to effect a change on some item representing an object.

I always got a kick out of speeding up any code, especially something in assembly language.

Probably over by now, but here in Chicago there used to be a huge demand for people to write programs to find the difference between stock options on the CBOT and stocks in NY and do massive trades. Something about the closer you are to the exchanges, and the faster your network connection and program is compared to others doing the same thing, the more money you can make.

I occasionally like to do proofs, such as what I did here for finding the values of a recursive sum:

https://en.wikipedia.org/wiki/User:Beneficii/recursum

It actually ends up pretty simple.

Further down, you can find a proof by induction of a simple statement equivalent to recursive sums.

Hi! To me nothing is more satisfying than solving a difficult problem. it doesn't matter to me if it has already been solved, so I like to work through problems in math olympiad collections. Actually I like to open up my old college physics text and work some random problems.

If you want unsolved problems, math is actually full of unsolved problems, if not mostly unsolved problems. Unfortunately you need to learn a lot of math to get to the unsolved problems. They live off in topology and exotic geometry. Here you aren't calculating things or trying to come up with a number, but prove something. Probably the most accessible ones are in number theory and combinatorics, and, like someone mentioned, graph theory.

I'd recommend looking at the popular books "The man who loved only numbers" by Paul Hoffman or "Count Down: The Race for Beautiful Solutions at the International Mathematical Olympiad" by Steve Olson to find the fun in math. Both are excellent and may inspire you to get out there and have some fun with math.

If you just want something to google on the internet and try out look up diophantine equations, or finding integer solutions to polynomial equations. It doesn't sound fun, now that I say it, but it is, or maybe look up recreational math at the khan academy.

Nothing. I know it's everywhere and all that, but really, math is just...you can't understand anything about it. I prefer to just take people's word for it.

Work on the Collatz Conjecture. A ten year old kid can understand it

See http://en.wikipedia.org/wiki/Collatz_conjecture

Here is another; Let S be a finite non-empty set. Let T(S) be the set of transitive relations on S. Find a formula which either equals or provides good upper and lower bounds for #T(S) the number of transitive relations on S.

There are plenty of unsolved mysteries in math. The more people discover, the more new problems people think of. It's like peeling an inside-out onion that gets bigger instead of smaller as you remove the layers.

Whether it's "fun" for you is a different matter. Unfortunately, most research mathematicians these days work on extremely esoteric problems. It might take years of education to understand just what the problems are, let alone their solutions. Most of those mathematicians are specialists, and maybe only a few hundred or so other mathematicians in the world understand their work. They may find their work fun, but only those few hundred can share their enthusiasm.

There are some exceptions. You mentioned prime numbers. There are actually many simply stated unsolved problems about primes. Lately there have been breakthroughs on the Twin Prime Conjecture. There is a lot of interesting research going on in integer factorization, elliptic curves, the discrete logarithm problem, and other problems related to cryptography.

Another exception, which someone else noted, is graph theory. This is a fun field with interesting, simple, unsolved problems and applications to computing.

I think that much interesting mathematics these days is generated by applications. Not traditional applications using differential equations (IMHO), but modern applications that involve optimization, stochastic processes, statistics, graph theory, etc.

Fields medalist Terence Tao seems to have done work in a wide spectrum of fun fields. I'm pretty sure he's done some work in compressed sensing. If you have some background in linear algebra and statistics, I'd recommend checking it out (statistics sounds boring, and it is often taught that way, but the more I learn, the more interesting I find it).

Hi,
I want to find something in mathematics that's fun to work on but also try to solve something that is still a mystery and unsolved to people. I enjoy prime numbers, geometry, algebra and trigonometry but there's no mystery. It's all been worked out. What I'm after is something that's interesting and hasn't been solved yet. Does anyone know of such a thing? All ideas welcome!

Wow. A lack of problems? It's all been figured?
I wish I could see that. haha!

Nevertheless, here's something I dare doing sometimes. Make a new mathematical object! Try to figure out what sort of things you want to be able to figure out with this new object and work your definitions so that it becomes operational. It's an endless fun frenzy of logic, creativity, maths and *cringes* sometimes, and only sometimes, philosophy.

I imagine the real fun starts when you figure out theorems out of your definitions and all that.

The last mathematical concept I've made was the concept of a pun space. Some whacky idea in order to explain the structure of a linguistic pun, and a generalization of a "pun" mechanism which works with other languages than written languages.
A friend of mine also came up with an analogy between physics' particles and natural and prime numbers, particularly by treating prime numbers as a made up particle, deducting its proprieties and so on. He was doing this with the aim of solving the Riemann Hypothesis in mind, if I recall. Of course, it's a little of a crazy approach.

Obviously, I won't get detailed here about the ideas you can think of. Either way, my point is, that if sometimes you'll allow your own inner mathematical censor to go on a vacation and get adventurous instead, you can not only find new mathematical problems, but create a whole new area of mathematics! (Warning: sometimes you might find out that other people have already thought of what you've created, but sometimes you find out they haven't! Keep trying.)

The fun never ends.