9. All horses are of the same colour

This puzzle has been attributed to the Hungarian mathematician George Pólya. It’s a cheeky “proof” by mathematical induction that all horses are of the same colour. (The proof does not say which colour.)

To prove that all horses are of the same colour, we prove something stronger, namely, that in any possible group of horses, all of the horses are of the same colour.

How does one prove this?

Well, suppose that, in any possible group of N horses, all of the horses are of the same colour. (N is an arbitrary positive integer, e.g., 7.)

Then it’s not hard to see that the same must be true of any possible group of N + 1 horses. For if you remove one horse from any such group, you will be left with a group of N horses, all of which, by supposition, are of the same colour. But you can easily replace the chosen horse back in the group and remove a different horse instead. You will now be left with a different group of N horses, all of which again are of the same colour.

Clearly this can only happen if all N + 1 horses are of the same colour.

So if the supposition is true of any group of N horses, it must also be true of any group of N + 1 horses.

Moreover, if N = 1, the supposition is obviously true, since any horse is the same as its own colour.

Q.E.D.