As an example let's have:
2
4
5
6
The guy with 6 has to guess first. Obviously he has no guaranteed success since his guesses are 1 3 or 6. At this point it's just 33.3%. There's no way to deduce his own number from the others if it's all random. He guesses 1.
Mr 5 (probably, if the rules are nonstupid) knows that his own hat cannot be 1. That leaves 3 5 and 6. He guesses 3.
Mr 4 knows that his own hat cannot be 1 or 3. That leaves 4 5 and 6. He might guess 4. If he knows that guesses 1 and 3 are incorrect, then he knows the other two bozos have 5 and 6, so he would be certain that his hat says 4.
(If the game ends when someone guesses correctly then that's enough: the game continuing is proof that a guess was incorrect)
From the start of the game, the first player to guess can only eliminate 3 of 6 guesses.
The second player can only eliminate 2 of 6 guesses, but can eliminate a third based on the first player's guess.
The third eliminates 1 of 6, but eliminates two based on the previous players' guesses.
Without some systematic strategy among the players I see no way for the second player to have guessed successfully. Again, does the first player WANT the second player to guess correctly? The problem statement SAYS that all other players CAN guarantee a correct guess but then asks for a strategy for the first player. Is this a strategy to HELP or HINDER the second guesser?? And without communicating among themselves, how does that even help?
There's another version of this where there are three players with coloured hats. There are two red and two blue; one for each, and then a spare which nobody can see. The players win if ANYONE guesses correctly which colour hat is on their head. The idea of this game is that it is trivial for the guy at the back if he sees two similarcoloured hats: "Hmm there's two blue in front of me, therefore mine is clearly red". It's not so easy if he sees one of each, but if the second guy sees that the first isn't immediately jumping to a conclusion, then he should know: "The guy in front of me is red and the guy behind me would easily have guessed that his was blue if mine was also red. Because he hasn't, that means he doesn't know, so mine must be blue!"
That one is clever and takes some thinking to understand. I just can't see how a similar strategy applies here...
Last fiddled with by TheMawn on 20150704 at 00:10
