Who's Shootin' Who?

Jan 29, 2007

RAY: This puzzler is mathematical in nature. Imagine if you will, three gentlemen, Mr. Black, Mr. Brown and Mr. White, who so detest each other that they decided to resolve their differences with pistols. It's kind of like a duel - only a three-way duel. And unlike the gunfights of the old West, where the participants would simultaneously draw their guns and shoot at each other, these three gentlemen have come up with a rather more civilized approach.

Mr. White is the worst shot of the three and hits his target one time out of three. Mr. Brown is twice as good and hits his target two times out of three. Mr. Black is deadly. He never misses. Whomever he shoots at is a goner.

To even the odds a bit, Mr. White is given first shot. Mr. Brown is next, if he's still alive. He's followed by Mr. Black, if he's still alive.

They will continue shooting like this, in this order, until two of them are dead.

Here's the question: Mr. White is the first shooter. Remember, he's the worst shot. At whom should he aim his first shot to maximize his chances of surviving?
RAY: Mr. White should aim at no one. He should miss.

TOM: What?

RAY: On purpose, and by doing so, he ensures that one of the better shooters, Mr. Black or Mr. Brown will be eliminated. Now neither of them is going to waste a shot at Mr. White, right? He's not that much of a threat.

TOM: Right.

RAY: So if he shoots wildly then Mr. Brown and Mr. Black have to go at each other. So after Mr. White misses, Mr. Brown whose shot it is now will obviously take aim at Mr. Black. If he misses then Mr. Black will return fire and kill Mr. Brown.

TOM: Yeah.

RAY: But no matter what it's now Mr. White's shot again. So by shooting into the air he guaranteed himself a second shot. He's still a lousy shot, one in three. But at least he gets to shoot first at the one and only opponent still standing.

TOM: Yeah.

RAY: Now for example if he shot at Mr. Brown and let's say, let's take the worst case scenario, let's say he shot at Mr. Brown and he killed him, you know lucky.

TOM: Yeah. He's a goner.

RAY: Because Mr. Black is going to shoot next and he's going to kill him. But by shooting and missing he makes sure that Mr. Brown takes out Mr. Black or vice versa, and then it's his shot again. If he had killed Mr. Brown for example, he'd have no chance at a second shot. This way he's got 100% chance.

TOM: So even though he has only got one chance out of three, he might actually kill somebody. And then he's done for.

RAY: He's probably a goner. We didn't ask how he could ensure survival, we just asked how he could improve his chances and that's how he could do it. So who's our winner?

TOM: The winner is Bill Spenz from Norman, Oklahoma. And for having his answer selected at random from among those thousands or three correct answers that we got, Bill is going to get a 26-dollar gift certificate to the Shameless Commerce Division at cartalk.com, where he can get a couple of our Car Talk dipstick mugs.

These are yellow mugs that appear from the outside to be one third filled with motor oil, and if you drank from one of these at the garage it probably would be filled with motor oil.

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