Prisoners and Hats

Mar 19, 2016

RAY: This was inspired by an email I got from Jim Gilford out in cyberspace. Here it is:

There's a penal colony on an island in the South Pacific. It's administered by a twisted prison warden who plays little mind games with the prisoners. He presents a challenge to the prisoners. If they solve it, they are set free. And if they don't, they're fed to the sharks. 

The warden says to five prisoners, "I'm going to stand you against the wall. One guy is going to face the wall with his hands and his toes touching the wall. The next guy is going to stand behind him about five feet away, and the next person behind him and so on. Each guy can see the back of the head of the guy in front, except for the last guy who can see everybody, and the front guy-- who can't see anybody.

"We're going to do this tomorrow," the warden says. "I want you to think about this overnight to see if you want to participate because, don't forget, if you lose -­" 

TOM: It's the sharks for you!

RAY: Right. And if you win, you get set free. Here's how it works. The warden says, "Starting at the back of the line, I'm going to place either a white hat or a black hat on each of your heads. I can put any hat I want on any of your heads. Your job is to identify the color of your hat correctly. There's one caveat. The guy at the back of the line can't communicate, because the only thing he can say is either 'black' or 'white.'"

Crusty, who's been on this island for 19 years for overcharging for valve jobs says, "I have a plan which will improve our odds beyond 50/50. However, we must draw straws."

The question is what is Crusty's plan and why must straws be drawn?
 
Answer: 

RAY: Say the guy closest to the wall has a black hat. The next guy has a black hat. The next guy has a black hat. The third guy has a white one. Then black, and then the last guy in line has a white hat. Here's what Crusty devised, and I think it was pretty smart.

The guy at the back of the line obviously can see all the other hats, but, of course, he can't see his. So he's going to guess, but he's not going to guess. He's going to declare his hat as either black or white based on what he sees. And they've determined in advance that if he sees an odd number of white hats in front of him, he's going to call out his hat as black. If he sees an even number of white hats in front of him, he's going to call his hat out as white. So he looks ahead and he sees what? An odd number of white hats in front of him. He sees one white hat, in fact. So he says, I have a black hat. You with me?

TOM: Yeah.

RAY: He goes to the sharks, but because he's declared his hat as black, the fellow who's directly in front of him knows that Crusty must have seen an odd number of white hats. Well, looking ahead, he sees what? An odd number of white hats --

TOM: Exactly!

RAY: Therefore, he knows he has a black hat. Similarly, the guy in front of him who has a white hat knows that the guy behind him declared his hat black, and because he had to have seen an odd number of white hats, he knows that his hat is white, because the only two hats he can see are black. Et cetera, et cetera. And everyone gets out, except for the last guy, whose chances are still 50/50. Pretty cute, huh? Do we have a winner?

TOM: The winner is Lieutenant Colonel Susan Robbins, United States Air Force, from Papillion, Nebraska.

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