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Unsolved mathematical problems
One of the great things about mathematics is that it generates a never-ending stream of questions: every time a question is answered, the answer tends to generate several new questions. As a result, there are - and always will be - a great many things that we mathematicians simply do not know.

Right now, the most famous of these problems (within mathematical circles, at least) is the Riemann Hypothesis, first posed by Bernhard Riemann in 1859. There's a $1 million prize for anybody who has a valid solution to that problem - so, answers on a postcard :P . However, that problem is rather difficult to explain to a non-mathematician - so, before tackling that problem, I think I should start with something more accessible!

The problem I've chosen is known as the "Collatz Conjecture" (after Lothar Collatz, who first asked the question in 1937). Sometimes, it's also known as the "3n+1 problem", because it involves multiplying numbers by 3 and adding 1:

Quote:Pick any positive integer (that is, any positive whole number). Then, do the following:

  • If your number is even, divide it by 2.
  • If your number is odd, multiply it by 3, and add 1.
  • Repeat the above steps until you get to 1 - and then stop.

For example, if you start with 3, this gives you the following sequence:
3, 10, 5, 16, 8, 4, 2, 1.

Or, if you start with 17, this gives you the following sequence:
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

The question is: do you always reach 1, no matter what number you pick? Or are there some numbers where this process keeps going indefinitely, without ever reaching 1?

Surprisingly, no mathematician knows the answer to this question! So far, it's resisted over 80 years' worth of attempts to solve it - and mathematicians aren't optimistic about finding a solution any time soon. We know (from brute-force computer calculations) it's true for every number less than 1020, but since there are infinitely many numbers to check, we'll never be able to "check every single number" like this :P !

So, what might the solution to these problems be? And, are there any other unsolved problems you'd like to mention?
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