Hats Off
Three mathematicians are applying for a job. There are five hats, three white, two black. They’re lined up, and a hat is placed on each. The first person in line cannot see any hat; the second in line sees only the hat of the person in front of him; the third person sees only the hats of the two people in front of her. The first person to correctly figure out what color hat he has gets the job; you guess wrong and you are killed. Assume these are INTELLIGENT mathematicians, and that they will do the logically correct thing at each stage — if something can be deduced, they will figure it out. After a long pause, the first person, who cannot see any hats, says he knows the color of his hat. What is the color, and how does he know?
What happens if there are 4 mathematicians, 4 white hats, and 3 black hats—can they all still be correct? What if there are n mathematicians, n white hats, and n – 1 black hats—does a general strategy still work? And if hats are assigned randomly, what’s the probability that everyone guesses correctly under an optimal strategy?
Thanks for sharing informative article.
Some interesting followups:
What happens if there are 4 mathematicians, 4 white hats, and 3 black hats?
What about if there are n mathematicians, n white hats, and n-1 black hats?
What is the probability, if there are n mathematicians, that each person in line gets the job (given that the hats are passed out randomly?)
Glad to hear!
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glad you’re enjoying; your email address didn’t work ./s
Very interesting, I got to admit it got me thinking for quite a time. 🙂
EXCELLENT question!
An interesting question would be the fairness of this job application since the other 2 mathematicians cannot ever determine their hat in the situation. If candidate “C” knows that he cannot determine his hat and that eventually A or B will, he might simply guess, or say he does not want the job and walk away. And let the others guess on their own.
Jason: correct (sjm1 AT williams.edu)
I’ll send a hint //s
I would like to know the answer and how the problem was solved.
Alex Irby: correct. Please email solns to me (sjm1 AT williams.edu) as I don’t want to post correct solns and spoil people’s fun. Thanks ..s
JerrySandusky: correct, not posting as it’s the soln ..s
glad you’re enjoying..s
Having read this I thought it was rather informative. I appreciate you finding the time and energy to put this informative article together. I once again find myself spending a significant amount of time both reading and leaving comments. But so what, it was still worth it!
xiaomilk: correct, well done — not posting as it’s the soln
to the person starting: Ok, let’s assume that person3 sees…. YOur logic from that point on is correct, but what if they don’t see what you claim? You’re essentially there, email me at [email protected] to chat more.
Ok, let’s assume that person3 sees 2 white hats infront. This means she doesnt know what hat she has, because it may be black or white. So she stays silent. Person2, who notices that person3 hasnt said anything, knows that he and person1 must have either a black&white hat between them, or 2 white hats. And since he can see that person1 has a white hat, then he doesnt know if he has a black or white hat (like, if he saw a black hat infront of him, then he knows that he’s got a white hat, but person1’s hat is white) so he stays silent. Person1, after seeing the other two staying silent, knows that there is only one possiblity that made both of them not sure of their colours: He has a white hat. So he answers, and gets the job.
email me at [email protected] with more details, as I think this answer is wrong. ..s
Black, the other two are white and can therefore not determine the color of their own hat.
email me at [email protected] for a hint
Is it that the he was wearing white? I’m not 100% sure how he had it….help?
To Anonymous (Submitted on 2011/11/19 at 12:49 pm): Correct! Not posting as it’s the soln.
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You can always email me at sjm1 AT williams.edu.
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Glad you’re enjoying the site. //s
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if this is 100 in a line, last sees all but his and no one ever sees their hat
eh, wouldnt the last person in line always be the first person to know his color since he can see all other 4 people’s hats? Or only can the first person in line make his/her guess? Im confused as to the way this question is laid out.
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The best thing to do for this one is draw all the possibilities, there are 7. Then work from the 3rd man forward and decide which ones required the “long pause” that let the first man to know.
Greetings all. We’re working on a teacher’s corner — when we have it ready, let me know if you want me to pass it along.
No one is able to see the extra hats, no one takes off hats, no one asks questions.
Here’s a hint: put yourself in the third person’s place. You see two hats in front of you. Go through all the cases of what you could see, and think if there are any cases where you would be able to deduce your hat color. if there are, since the third person doesn’t speak then those cases cannot happen.
Can the first person see the hat’s that are left over?
If the answer is yes, we believe the first person sees that two black hats are left and is able to deduce that the three white hats are on their heads….
OR
Is there any rule that says they can’t take their hat off and look at it..maybe the other two were concentrating so hard on trying to solve it based on the hats they COULD see, and the 1st mathemetician recognized there was no rule saying they could not take their hat off.
OR
Can they ask questions of each other (which I don’t think makes sense, because why would you help someone else figure it out if you want the job….my students are nicer and think they would want to help eachother to stay alive…silly students, they don’t know how hard jobs are to find these days). Can they ask questions of people not in line? Are their people not in line…
please help. we need to go to lunch now…
oh…p.s. my students want to know if everyone else envisions top hats when solving this problem…because we all did!
Thanks! Ms. Lippman and Brilliant Students
Sure, but let me try a hint first. Imagine the third person saw two black hats. If this were the case, would they know what they have? Keep pushing this logic forward to eliminate what the hat possibilities are. If this doesn’t help, email me at sjm1 AT williams.edu for a solution.
Hello. Could you send me answer for this riddle? I dont know how to do this… 🙂 If it is some kind of secret or you want me to think more I understand. Thank you in advance.