No assumptions needed.
Tom makes a statement.
Dick then says, "In that case, . . . " i.e. given Tom’s information, I conclude . . .
Harry “considers what both had to say”. In other words he uses the information that both have given.
The line of reasoning is quite linear. I understand your concern about my logic in determining why all three hats must be white, but this is similar to the famous “Three dots problem”.
Well, figure the men are given one minute to think. When a buzzer sounds, they must, if they can, announce their hat colour.
Minute 1: If there are two green hats in play, the man with the white hat will announce.
Minute 2: If no-one has announced on the first round and there is one green hat in play, the two men with white hats will announce.
Minute 3: If no-one has announced on the first or second round, then there are no green hats in play and all three men will announce.
Fair enough? I guess the man/men in the green hats get screwed because there’s no way to be the first (or tied for first) to figure their hats out. At best, they can announce only after someone announces their (white) hat(s).
In the version I’ve heard, Harry is blind, and yet he still knows he’s wearing a white hat. The reasons have been stated already, but with a lot of noise, so let me try again:
Tom cannot guess the color of his hat, so we know he doesn’t see the two green hats on Dick and Harry. That’s all we know at this point.
Dick cannot guess the color of his hat, yet we know if he saw a green hat on Harry, he would know he (Dick) was wearing white. Why? Because they aren’t both wearing green.
Since we know Dick didn’t see a green hat on Harry, then we know Harry must be wearing a white hat. Harry also knows this without even having to look at Tom and Dick.
SaintCad, I see what you mean but I agree with Tyrrell that it’s only valid if you assume that everyone can speak at the same time. I understood the setup to mean that the three men are only allowed to speak in the order given. This makes a difference as the following two cases show:
Case 1: T, D and H are all free to speak whenever they like (and let’s assume that they are all equally good reasoners).
WGW
Tom sees one white and one green hat. So does Harry. There is a brief pause, during which Dick says nothing.
Tom and Harry both immediately conclude that they are wearing white hats, since they each can see one green and nobody has hollered out “I’ve got a white hat!” immediately (meaning that nobody saw two green hats).
Case 2: T, D and H must make their statements in order, one at a time.
WGW
Tom sees one white and one green hat. So does Harry. Here’s the kicker - neither Dick nor Harry can say anything, so Tom can’t know his own hat color. Silence from D and/or H doesn’t mean anything, it’s not a piece of information for Tom to use.
Why did you make that assumption? There is nothing in the puzzle description that implies that there is any order.
Well, then Tom and Dick are the slow ones, for speaking when they should be thinking.
Sure there is. Tom says X. THEN Dick says “In that case, Y”, and FINALLY Harry considers what Tom and Dick have both said and comes up with Z. This implies a particular order and that they aren’t all speaking at once.
This is what makes this puzzle, as given, a little different from similar setups such as the “All the blue-eyed people have to commit suicide” problem, where everyone reasons at the same speed and decisions are made simultaneously on a “once per day” basis.
But the order is not forced upon the players by an outside soirce i.e. nothing says Tom has to be the first to speak then Dick must speak second then Harry has to be the last. Each speaks when they reach a conclusion . . . unless you’re arguing that there is no free will.
Then define the rules rather than argue about what the rules aren’t. There have been discussions of how the game would play out given certain assumptions and as described in the OP, Tom and Dick apparantly voluntarily (and all things considered, stupidly) tipped their hands by revealing their ignorance. In contrast, if the three men operated with perfect logic, none would ever say “I don’t know”, as all this does is give information to the other players.