A couple of years ago I wrote an article on the famous four colouring theorem. (For those who missed it, the basic idea is that any map drawn on a flat plane will be able to be coloured such that no two neighbouring regions have the same colour, using only four colours). Although this problem has been solved, with a proof being completed nearly forty years ago now, it is a controversial problem in that the only known proof of this theorem is a brute force calculation by a computer, rather than an elegant a sophisticated bit of mathematical logic. As such some people still work on this problem, with the hope of eventually finding a simple proof.

However that is not the topic of today's article. Instead I thought I would present three very similar problems in mathematics for which there is no known proof - brute force or otherwise. In each case the conjecture is fairly simple to state and to understand, but is as yet unproven. And they are also the exact sorts of problems that amateur mathematicians can attack and enjoy.

So without further ado, here they are...

The Total Colouring Conjecture: If the four colouring conjecture is about maps of countries, then the total colouring conjecture is about maps of cities and highways. If you have a map (or a graph in mathematics terms) of cities and highways that connect the cities, then how many colours are required to colour in every city and every highway, such that no highway has the same colour as a city on either of its endpoints and no two cities connected by a highway have the same colour. Clearly the number of colours cannot be less than the maximum number of highways connected to any city (also known as the degree of the graph). In fact we know that it will always be at least one greater than the degree of the graph. The conjecture is that it will never be more than two greater, but as of this writing that has not been proven.

The Hadwiger-Nelson Problem:
This is one of the simplest and yet most difficult conjectures. It is remarkably simple to state, and yet so difficult to prove that as yet no one has. Suppose you divide the two-dimensional plane into a large number of regions, and assign a colour to each region. However there is one condition on both the drawing of the regions and on the colouring. This time you need to end up with a map that has no two points in it that are exactly one unit of distance apart and have the same colour. Or another way of stating this problem is that each and every equilateral triangle whose sides have length one must have three different colours on its vertices. The question is, how many colours are required to completely colour the plane?

Conway's Thrackle Conjecture: Although not a colouring conjecture, this one is still fun to play around with. Draw a set of points on a piece of paper. Now draw lines connecting the points, with the one condition being that each line you draw must intersect every previous line you have drawn, either by both being connected on one end to the same point, or by crossing the previous line exactly once. The conjecture is that it can never have more lines than the number of points that you started with, but this has not yet been proven.

And so there you have three simple but interesting problems in mathematics. Like the four colouring theorem, each can be stated in simple terms, each can be studied by playing around with diagrams, and yet in each case the answer is not known. 

So I hope that each of you will play with these problems, and maybe even find a solution to one or more of them!