Horses of colour
Can maths be fun?
You probably fondly remember your days of algebra and err, that other thing, begins with 'c', I think. Maths and humour do mix, though only rarely, and here is an example.
Recall that the procedure called mathematical induction works as follows.
Step one is to show that the general statement is true in the case n = 1.
Step two is to show that if it holds for n = k then it can be shown to hold for n = k + 1.
Step three is to conclude that it holds for all n.
That was the unfunny part. Now we prove that all horses are the same colour.
Step 1. If you look at a single horse then it is by definition the same colour as itself. So far so good. The case where n = 1 is secure.
Step 2. Assume that any collection of k horses is the same colour. We will show that it follows from this assumption that any collection of k + 1 horses is also the same colour.
Proof: Take any group of k horses. These are all the same colour, by our hypothesis. Now take away one of these horses, stable it next door so it won't gallop away. Bring in a fresh horse from somewhere else (a pony won't do). Now we again have a collection of k horses and therefore they are all the same colour, since this is the hypothesis.
But what about the horse waiting next door? Surely it is the same colour as the original k-1 horses? Yes, it is, because when it was part of the original group of k happy horses it was the same colour as all the others. It still is (assuming it wasn't sprayed pink on the way to the stable). The ring-in is also the same colour (by hypothesis), so all the horses so far mentioned, ie k+1 of them, are the same colour.
This is what we were trying to prove.
Step 3. It follows that all horses are the same colour. (But what colour? you ask.)
Question: where is the logical slip in the above proof?
Solution