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Logic Level 3

A prison warden plays the following game with 4 male prisoners, who are all eager to escape.

Three of them are standing on steps looking straight ahead, such that the 1 $^\text{st}$ one at the top can see both the 2 $^\text{nd}$ and 3 $^\text{rd},$ and the 2 $^\text{nd}$ in the middle can see only the 3 $^\text{rd}$ at the bottom who can see nobody. The 4 $^\text{th}$ prisoner is in a room, totally separated from the other three. But they can all hear the voices of others.

The prisoners know that there are 2 black and 2 white hats available for them each to wear, but they do not know their own hat color. The 1 $^\text{st}$ is wearing a black hat, the 2 $^\text{nd}$ is wearing a white hat, the 3 $^\text{rd}$ is wearing a black hat, and the 4 $^\text{th}$ is wearing a white hat.

The prison warden promised that the first to shout his color correctly will be released, otherwise he will be shot. Who will be the first to shout his hat color?

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3^\text{rd}

4^\text{th}

1^\text{st}

2^\text{nd}

7 solutions

Abel Chen
Jul 8, 2014

The 3rd and 4th prisoner cannot logical deduce anything because they have no information.

The 1st prisoner sees the hats of the 2nd and 3rd prisoner. If he sees that the two prisoners in front of him are wearing the same colored hats, he will know what color he is wearing, since he knows that there are two hats of each color. But since they are different colors he cannot be sure whether he is wearing black or white.

Since the 1st prisoner doesn't announce his hat's color immediately, the 2nd prisoner knows that he is wearing a different hat from the 3rd prisoner, the one in front of him. Since prisoner 3 is wearing black, prisoner 2 will know he is wearing white, and shout his hat's color first.

Since no prisoners are allowed to turn their head or eyes and no one can see the 4th prisoner in his room..no one knows that he turned his eyes...and thus he'll shout first ;)

Pranay Chris - 6 years, 10 months ago

See my logic was hed take his hat off

Samantha Wimmer - 5 years, 5 months ago

That's what I thought! If no one can see him then no one will know if he looks at his hat's color.

Danielle P - 5 years, 2 months ago

The prisoners can just remove his hat and check the color. After-all they are only not allowed to turn their heads or eyes.

Ranganath Govardhanam - 6 years, 9 months ago

Assuming u r the third person, and u r looking at the first 2 person and could not determine the colour of the hat he is wearing. Thus, he will remain silent. Using this logic, the second person guess that since the third person did not shout out the answer, he then must have a hat that is of a different colour from the first person. However, the poor first person could not see anyone's hat and remain clueless as person 3.

wong mark - 6 years, 7 months ago

The problem did not state that the prisoners had to say their hat colour in the order of 1st prisoner, 2nd prisoner... How are we supposed to know that the first prisoner has the first chance to either say something or not? I figured anyone could speak at any time so there was no was for anyone to know

Kyle McLean - 5 years, 4 months ago

That dumb dumb

Leon Wang - 5 years, 3 months ago

In response to Abel Chen Only problem is the second prisoner "can only see the one in front of him" he could be facing prisoner 1 or 3

John Zickmund - 4 years, 11 months ago

This question is flawed based on its very premise. The fourth man would know first.

He is in a room by himself, thus no one is present to prevent him from breaking the rules which, as a criminal, he is prone to do. He would simply look at the hat's color and shout it.

Brian Mitchell - 4 years, 9 months ago

That isn't a premise of the problem. Though nuance is often lacking in applied analysis, there is no data provided that allows one to infer more from the problem than what is stated--even, if you want to get technical, that the prisoners are even human.

Thus, your objection MAY be valid on the presupposition that sight is not the only way for each of the prisoners to know something about the other prisoners (via telepathy, repeated game equilibria, etc); but presupposing a breach of the rules based upon being "a criminal" is bad inference... not to mention terrifyingly horrible policy!

Joshua Nesseth - 4 years, 3 months ago

Or, if he does know the color, and does not say it, the second prisoner would speak up, and get killed. It's pretty much "do you trust this first guy to not get you killed."

Josiah Kiok - 4 years, 4 months ago

Chew-Seong Cheong
Jul 14, 2014

The prisoners are all smart. The 1st prisoner has the most information and since he does not shout his hat's color, meaning the two that he can see of not the same color. Knowing this fact, the 2nd prisoner knows that he is wearing a hat different color from the 3rd one in front of him and is the first to shout the color of his hat.

Actually, your comment about the 2nd prisoner being the only one able to guess the color would be incorrect. If he could deduce that he had a different color than the 3rd from the silence of the 1st, the 3rd would be able to guess the same thing and shout his color hat second. They would be the only ones able to guess.

Melissa Bonvi - 5 years, 5 months ago

Yes, you are right.

Chew-Seong Cheong - 5 years, 5 months ago

Arnab Das
Sep 7, 2014

This problem can also be solved with the the help of basic probability. The 4th prisoner is absolutely isolated so there is no way he can predict the colour of his hat. He has absolutely no information. Same is the case with prisoner 3 as there is no body above him .

Now let us consider prisoner 1. He can see that prisoner2 is wearing a white hat and prisoner 3 is wearing a black hat. So there a two hats left out of which one is black other is white out of which one is his.There is a 50-50 chance . BUT For prisoner 2 , he can see that prisoner 3 is wearing a black hat. So the left out hats are 2 white and one black out of which one is his. So probability of white is 0.66 (2/3) for white and 0.33 for black(1/3).

Hence the answer is prisoner 2

Tom Dufall
Jul 26, 2014

The 2nd prisoner. The 2nd prisoner knows that if he and the 3rd prisoner had the same coloured hat, the 1st prisoner would call out, since he can see both of them and would know that his is the opposite colour (since there are 2 of each hat.) When the 1st prisoner doesn't call out, the 2nd prisoner realises that this means that he (prisoner 2) and prisoner 3 (in front of him) have different colour hats, and so he looks at the colour of the hat of the prisoner in front of him (the 3rd) and calls out the opposite hat colour! [Edit: I numbered prisoners 1-3 in the wrong order, so updated to correct this]

Good explaination! :) East to understand!

Elias Adler - 4 years, 11 months ago

Poonayu Sharma
Jul 7, 2014

The question of either 3 or 4 saying their hat colour doesnt rise.....

If prisoner 1 cannot make out the colour of hat he has, then 2 should be sure that he was wearing the hat of the other colour than black because if 1 sees 2 persons ahead of him wearing different coloured hats, he only has a 50 percent chance to say his hat colour. So if 3 stands confused and silent for some moments ....2 shall know he is wearing he hat of the opposite colour of the one in his front ..

Yeah because the second one know the first one cannot answer because the first doesn't sure what he wears.. so that's mean the 2nd and the 3rd wearing a different hat

Ali Akbar - 6 years, 11 months ago

No you are wrong how does 2 know that he is wearing opposite color of prisoner before him since no one knows their own color? 2 shouted first because he know 3and 4 cannot make answer so he waited for 1st prisoner's answer who has high priority of answering by seeing 2 prisoner's hat infront of him.. As he didnt answered instantly 2 became to know that 1 is confused just because both prisoner infront of him wearing different color so 2 shouted the color of hat opposite to 3..

Sha Beer - 6 years, 11 months ago

John Zickmund
Jun 21, 2016

1st prisoner can see two hats of different colors therefore 50/50 chance not enough to guess. 2nd prisoner can only see one hat but that is enough to give himself a 66.6/33.3 chance of his hat being the opposite color than the prisoner he can see. This is the prisoner with the best chance therefore first to guess even if he is wrong. 3rd and 4th can't see anyone so they have a 50/50 chance. The 2nd prisoner guesses first.

Sharad Lal Amatya
Jul 19, 2015

A classical example of lateral thinking.

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