the four colour theorem

How many colours does it take to colour a map? The title probably gives you a hint.

When I was a kid (no, I don’t know how old exactly… around 10), my dad set me and my brothers a challenge. Could we create a map which required more than four colours to colour it in? Here are the rules:

– no two adjacent “countries” can have the same colour,
– if two “countries” meet only in a point then they are not adjacent, and
– each “country” must be completely connected (Alaska, for instance, couldn’t count as part of the USA)

At the time I had no idea that this was a world famous mathematical concept, I just thought it was a frustrating problem. We didn’t find a map requiring 5 colours and, indeed, we couldn’t have. It was proved in 1976 that 4 is the maximum number of colours needed and, importantly, it was proved by using a computer.

In 1976 this was revolutionary, groundbreaking and contentious. Lots of mathematicians refused to accept the proof because it couldn’t be checked by hand. It has since been proven by a computer program we know to be reliable but it was a point of argument among mathematicians for many years.

 

Research in mathematics?

I am asked, regularly, how there can still be things to research in maths. How can there be new maths? Two is still two, right? What could there possibly be to discover?

So, for those of you who wonder how the career “mathematician” still exists, or what I could possibly spend three years researching, I have decided to create a weekly feature of famous unsolved mathematical problems/current research in mathematics.

The first thing that jumps to my mind is Goldbach’s Conjecture. Goldbach conjectured*  in the 1700s that any positive even number (>2) is equal to the sum of two primes. This is easy to see for the first few examples: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5 and so on. Using the amazing powers of computers we know that this conjecture is true up to ridiculously high numbers. But no one has ever proved that it must always be true.

This is such a great problem because it can be understood by anyone who knows what a prime is but it’s obviously incredibly difficult to solve.

Personal note: I was set this question as ‘homework’ by the head of the Adelaide Uni Maths Society a few years ago. He knew that it was a famous unsolved conjecture but I did not. I spent several hours trying to solve it but was sadly unsuccessful. My dad laughed when I told him what I was working on. He knew what it was too. Now you all know and can set the problem for your mathematically inclined friends.

*Actually he conjectured some very similar things, one of which turns out to be equivalent. Go look it up on wikipedia.