Unsolved mathematical problems - Printable Version

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Unsolved mathematical problems - Kyng - 06-10-2019

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 . 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 !

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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RE: Unsolved mathematical problems - Kyng - 07-30-2022


Here's a video that lists three geometry problems that are easy to state, but hard to solve (and indeed, no mathematicians have solved them yet ). Here they are:

• The Happy-Ending Problem. This problem was given its name by Paul Erdős, because two of his fellow mathematicians (George Szekeres and Esther Klein) met while working on it, and later got married - but we don't yet have an ending for the problem itself .
  
  If you draw five points on a sheet of paper, such that no three form a straight line, then four of them will form a convex quadrilateral (that is, a quadrilateral where all angles are less than 180 degrees). If you have 9 points, then five of them will make a convex pentagon - and if you have 17 points, then six of them will make a convex hexagon. But how many do you need for a convex heptagon, octagon, or any other n-gon? We don't know... all three of these results fit the pattern "for n sides, you need 2^n-2 + 1 points" - and Erdős and Szekeres conjectured that this might be true for any n >= 3 - but nobody's been able to prove it.
  
  
• The Inscribed Square Problem - On a sheet of paper, draw a loop of any shape you want, as long as it doesn't cross over itself (so, it's what mathematicians call a "Jordan curve"). Now, can you find four points on your loop that form a perfect square?
  
  Mathematicians have tried this with loads of loops, and it's worked for every loop they've tried so far - but they haven't been able to prove that it works for every loop.
  
  
• The Perfect Cuboid problem - Is there a cuboid where the three sides, the three diagonals across each shape, and the diagonal through the middle of the cuboid, are all integers? (So, on this diagram, is there a cuboid where the lengths a, b, c, d, e, f, and g are all whole numbers?)
  
  Mathematicians have had lots of near-misses, where six of these lengths are integers - but they haven't found a cuboid where all seven of them are. On the other hand, they haven't been able to prove that such a cuboid doesn't exist .

If you have a solution to any of these, then answers on a postcard!

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