Vinnie Goombatz and the 247 Arm Wrestlers

Feb 22, 2010

RAY: This is stolen somewhat from an e-mail I got a long time ago. Here we go. Our erstwhile companion and chief mechanic, Vinnie Goombatz, being renowned for his prowess in arm wrestling, is asked to set up a tournament at the local watering hole where he goes and gets stewed every night. It's to be a single elimination tournament, i.e., once you lose, you're out. No ties allowed. This is arm wrestling; you can't have a tie in arm wrestling, right?

TOM: Yeah.

RAY: To his horror, 247 people have signed up for this tournament, and the barkeeper wants to know how many bouts have to be fought. Figuring a bout takes about five minutes, he wants to know at what time he should start the event so that it will conclude before closing time.

So Vinnie is in a tizzy now, because he's thinking about, Oh, I gotta set up a branching tree, count all the branches, and since he can't count much beyond 14, he's in a tizzy.

Fortunately, there's a little guy sitting next to Vinnie at the bar, and the guy says, "I know the answer." Vinnie says, "What are you, some kind of genius or sump'm?" The guy says, "No, but there is a simple reasoning process which will allow you to instantly know how many bouts have to be fought."

The question is: If there are 247 people that signed up, how many bouts will there have to be in order to determine one winner?

TOM: With a single-round elimination.

RAY: And show your work. And, by the way when the bout starts, both hands of the clock are 180 degrees apart.
RAY: The question is, how many bouts have to be fought in order to determine one winner? One winner. So, you start off with 247 people. Divide that group in half, right? Half of them are gonna wrestle the other half. Then you're gonna lose half of those people.

TOM: Right.

RAY: And that half is gonna wrestle, right? You could go and do all this, but there's a simpler way to figure it out.

TOM: There is?

RAY: According to John LaTorre, who sent this to us, he claims that Albert Einstein used this as an example of elegant reasoning. That is, reaching a conclusion in the fewest number of steps in his math lectures. And here's the answer. Since you can't have any ties, every bout must have a winner and a loser. And since the thing is a single-elimination, everyone will lose once, and only once, except for whom?

TOM: One guy! The winner.

RAY: Therefore, how many losers are we gonna have?

TOM: Two hundred and forty-six.

RAY: How many matches are we gonna have?

TOM: Two hundred and forty-six.

RAY: Do we have a winner?

TOM: Yeah. The winner this week is Jack Samuel from Cave Creek, Arizona!

RAY: Congratulations Jack!

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