There are 6 ex-prisoners who have recently escaped from a hat-puzzle-prison by beating the warden at his own game. While enjoying their new-found freedom, they happen upon a travelling milliner who has a big sign on his cart - "6 hats for the price of 1 if you can solve my hat puzzle!"
To the ex-prisoners, this sort of puzzle is 'old hat', so they jump at the chance, and rush to the milliner to take advantage of the deal.
Before they get a chance to introduce themselves, the milliner hails them in a booming voice: "Welcome, welcome. I can see you ladies have fine taste in hats - and you appear to be great thinkers as well. What luck that I just happen to have in a new shipment of 'Thinking Hats' from the great de Bono himself. They come in six colours - white, red, black, yellow, green, and blue - and they would match your complexions perfectly! And I know what you're thinking - I'll let you have 6 for the price of 1, if only you can solve my hat puzzle.
The Milliner wrote:I'll ask you each to close your eyes, and stand in a circle. On each head I'll put a thinking hat, of a colour that I choose myself. I'll choose from amongst the 6 colours available, but I won't necessarily use all 6 colours - there may be some repeats. Once you open your eyes, you'll be able to see the hats of all of your fine companions, but of course you won't be able to see what colour your own hat is. You then each get one chance to tell me (in secret) what you think your hat colour is. As you can tell, I'm a great observer - I already know each of your hat sizes, for example - so I'll certainly know if you communicate with each other and give the game away!
Now, I'm a reasonable fellow; I don't expect everyone to guess their hat colour correctly. If at least one of you correctly guesses your hat colour, then you win. You can't say fairer than that, can you? Well, I can see from your faces that you're already sold on these hats - so let me know when you're ready to take up this amazing offer!
The ex-prisoners step to the side of the road, and start to confer, since they won't get a chance after accepting the deal. What strategy should they use to maximise their chance of getting at least one correct guess?
(I know what you're thinking - there's a decent chance they'd get it right even if they just guess randomly. However, unbeknownst to the ex-prisoners, the hats are rather overpriced, so the milliner will still make money if they get it right anyway. But, then, he has to make a living somehow!)
Bonus puzzle: Generalise the solution to n hats (so n prisoners, n different hat colours available.)