# The “What Color is Your Hat” Brainteaser

A teacher puts 10 hats of either red or blue on 10 students.  Each 1 can see the hats on all the other 9 students but not his or her own.  The teacher tells them that at least 1 of the hats is blue and they then ask the students to write down the color of the hat that they are wearing if they are sure about it.  Or, the student can write down, “Don’t know.”  Everyone reveals their answer at the same time and all the right, “don’t know.”  On the 2nd day, they gather in the teacher puts on the same hats on each student.  Each 1 of them has to think about the color of their hat again.  This time, no one can still figure out the color of their hat.  Everyone writes down, “don’t know.”

This game repeats itself on the 3rd day, 4th day until the 9th day and still, no one can figure it out.  However, on the 10th day, everyone writes down the correct color of their hat.  What happened?  What is the color that they wrote down?  The assumptions here are that throughout the 10 days, the students do not communicate with one another and they assume that everyone is smart in those that everyone else is smart as well.  That’s the problem we have to solve.

The correct answer is that they are all wearing blue hats.  There is proof to it.  Follow the logic here.  Credit for the answer goes to Pratik Pradar.

If there are k number of blue hats and, but all of them can tell at time k that they are wearing blue hats.  This can be proved by induction.  If there is only one blue hat, it would seem that no one has a blue hat.  But there has to be someone with the blue hat so it must be him.  He gets it in day one.

If there are 2 blue hats, they will think there is only one blue hat, then the person would’ve said on the 1st day, so it must be a blue hat.  And so on.  On the kth day, all K people realize that they have blue hats.  Since here K equals 10, there are 10 people with blue hats.

Ⓒ The Big Game Hunter, Inc., Asheville, NC 2020

ABOUT JEFF ALTMAN, THE BIG GAME HUNTER

Jeff Altman, The Big Game Hunter is a coach who worked as a recruiter for what seems like one hundred years. His work involves career coaching, all as well as executive job search coaching, job coaching, and interview coaching. He is the host of “No BS Job Search Advice Radio,” the #1 podcast in iTunes for job search with more than 1900 episodes, and is a member of The Forbes Coaches Council.

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