When the Four Colour Theorem (FCT) was finally ‘proved’ in 1976, it upset a lot of mathematicians. It was the first significant mathematical concept to be proved with a good deal of help from a computer and, for many, that didn’t make it a real proof. Although we’re largely (maybe not entirely) OK with it now, the objections at the time weren’t just theorists’ snobbery. At the heart of it all were some fundamental questions about the role a computer could or should play in formal logic.
Essentially, the FCT says that the maximum number of different colours needed to colour a map, so that no bordering countries are the same colour, is four. (Colours can touch at a point but not at an edge.) It’s easy to show that five will always do the trick and, in fact, most normal maps only need three. However, certain types of map certainly seemed to need four so was four always enough? Continue reading
That looks like a hopelessly vague question, and it is unless we’re prepared to clarify it a bit. On the other hand, we already know there are some impossible problems so surely there are some that are just hard? Again, we’ll need to work out what on earth we’re talking about here. Let’s start with what we actually mean by a problem in a computational sense …
(Be warned: There are one or two simplifications and liberties with precision in what follows; it’s well-intentioned but may upset the purist.)
Well, actually, even that isn’t simple and there’s no absolute agreement on what a good definition would be. (We’ve seen previously that mathematicians and computer scientists don’t always see eye-to-eye.) It’s cheating a bit but it’s probably easier to give examples and this should work well enough for us. At least in the context of computing, these are all valid problems:
- Calculate 2 x 4 + 9 – 3
- If 5 – x = 2 what’s x?
- Find the largest from 5, 7, 1, 4, 8, 5, 2, 4, 8, 5, 2, 6, 7, 7, 3, 3, 2, 4, 3, 6, 7, 7, 6, 5, 4
- Sort 25, 44, 66, 72, 12, 45, 56, 90, 45, 69, 11, 10, 12, 42, 88 into ascending order
- Arrange 1, 2, 3, 4, 5, 6, 7, 8, 9 into a magic square
- What’s the best way to get to Paris?