# Logic - 3 Cannibals Riddle Reduced to 2 Steps

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Cannibals ambush a safari in the jungle and capture three men. The cannibals give the men a single chance to escape uneaten. The captives are lined up in order of height, and are tied to stakes. The man in the rear can see the backs of his two friends, the man in the middle can see the back of the man in front, and the man in front cannot see anyone. The cannibals show the men five hats. Three of the hats are black and two of the hats are white. Blindfolds are then placed over each man's eyes and a hat is placed on each man's head. The two hats left over are hidden. The blindfolds are then removed and it is said to the men that if one of them can guess what color hat he is wearing they can all leave unharmed. The man in the rear who can see both of his friends' hats but not his own says, "I don't know". The middle man who can see the hat of the man in front, but not his own says, "I don't know". The front man who cannot see ANYBODY'S hat says "I know!" How did he know the color of his hat and what color was it?

The answer is: The man in front knew he was wearing a black hat because he knew the first man did not see two white hats and he knew that the second man did not see one white hat because if he saw a white hat, the second man would have known that his hat was black from hearing the first man's statement.

However I think it can be simplified and would like to prove my reasoning so you guys can tell me why I'm wrong / right

Man in back: I don't know proves that the hats in front of me cannot be both white, because the only possible way he could know would be if both hats he saw were black, therefore one is black and one is white.

Man in middle Knowing this information, the man in the middle thinks the following things if he sees either scenario:

He sees a black hat on the man in front of him: His should be able to conclude that his hat is white because there is a hat of each color, so if the one in front of him is black, his must be white

He sees a white hat on the man in front of him: He is able to conclude that his hat is black because since there is a hat of each color, if the one in front of him is black his own must be white

In the answer given, it's required to hear the third man before we can know the answer, but why isn't my above solution correct? which solves it in only 2 steps.

• Why can't the man in back see two black hats? He still wouldn't know if he had the third black hat or if he had a white hat? – barrycarter Oct 18 '14 at 0:46
• Because the riddle specificially states that they are able to figure out who is wearing what hats, and it wouldn't be possible if he saw two black hats. – Keatinge Oct 18 '14 at 1:06
• "the only possible way he could know would be if both hats he saw were black". I'm pretty sure that's a typo. The only way he could know is if both hats were white (he'd know he had a black hat then). – barrycarter Oct 18 '14 at 4:07
• What if all prisoners wear black hat? – NEO Sep 5 '17 at 8:18

you write:

He sees a black hat on the man in front of him: His should be able to conclude that his hat is white because there is a hat of each color, so if the one in front of him is black, his must be white

That is wrong; assuming the first (to answer) prisoner gave the correct answer, the second prisoner can only conclude there can't be two white hats, and since he said he didn't know what color was the hat he was wearing, it implies the prisoner in front was in fact wearing a black hat.

That is how the prisoner in front knew what color was the hat he was wearing.

It can be done in one step. The first man sees two white hats and says that he knows.

But, you say, the problem says that the first man says "I don't know"?

Exactly. The problem's definition is also specified by the second man saying "I don't know."

Therefore, if you can break the definition of the problem for your two steps, I can break the definition of the problem for my one step.

...

The riddle is solvable without further assumptions, and it takes three steps.

• If the man in the back saw two white hats in front of him, he knew, that he would wear a black hat. But he states, that he does not know: The two hats in front of him are not both white.

• So the the man in the middle knows, that he and the guy in front of him are either wearing both black hats or mixed-coloured hats. If the guy in front of him wore a white hat, the guy in the middle would know, that it is the mixed-hat-scenario. Then he could conclude, that he wears a black hat. But obviously he cannot conclude that.

• So the front-man knows that he is not wearing a white hat. He must be wearing a black hat.