RAY: Hi, we're back. You're listening to us, Click and Clack, the Tappet Brothers, and we're here to talk about cars, car repair, and, uh, the answer to last week's Puzzler. I guess you'd have to call this a mathematical Puzzler, and the one thing I liked about it was it was brief.
TOM: It was brief and it left little room for obfuscation. And it didn't need any obfuscation.
RAY: All right. Here it is.
TOM: Go ahead.
RAY: I gave Tommy a thousand $1 bills, and his assignment was, first, come up with 10 envelopes, and once he did that he had to figure out a way to put various numbers of dollar bills in those 10 envelopes so that no matter what amount of money I asked him for, he could hand me some combination of envelopes and always be sure of giving me the exact right amount. The question is: How did he do it?
TOM: That's the question? I thought the question was, What was going on in your mind that you gave me 1,000 $1 bills?
RAY: I could've given a hint last week and said one of the envelopes has $489 in it. I could have given that hint.
TOM: Really?
RAY: Well, yeah, and the other envelopes...the other nine have $1...
TOM: I like that.
RAY: $2.
TOM: Yeah.
RAY: Four dollars, $8, $16, $32, $64, $128, and the ninth envelope has $256. If you add those up -- 256, 128, 64, 32 -- you come up with 511, because in base 2, the next number would be...
TOM: 512.
RAY: 512, OK? Two to the tenth would be 512, but he couldn't put 512 because you don't have it in there.
TOM: No.
RAY: So you could put 489. So you can get any possible number between one and 511 by using the first nine envelopes, and then anything beyond 511 up to a 1,000 using 489 plus one gives you 490, 490 plus two gives you, and da-da-da.
TOM: Man.
RAY: Pretty good, hunh? Give me my thousand bucks back. It was only a loan.
TOM: Give me my envelopes back.
RAY: A further demonstration of the power...
TOM: Of two.
RAY: Of two.
TOM: Man.
RAY: You can't do it unless the number is two. There you go, see? The power of two. Right?
TOM: They just won't leave you alone. I'm almost feeling sorry now. I'm going to have to stick up for my little brother. Lay off the guy!
RAY: I know, relentless. Everybody too.
TOM: I told him to do it. We have a winner. The winner is Betty Benson from Houston, Texas, and for having her answer selected at random from the wastebasket full of correct answers that we did get, Betty is going to get a $25 gift certificate to the Shameless Commerce Division, which is at the Car Talk section of cars.com, and with that $25 gift certificate, Betty can get 1.39275660167ths of a Car Talk T-shirt that says "Don't Blame Me, I Voted for Click and Clack." There's one of these in the Smithsonian, you know.
RAY: There is?
TOM: Yeah, Smithsonian Transmission Rebuilders on Landsdowne Street.
RAY: Good luck, Betty. Congratulations. Anyway, we'll have a new Puzzler coming up in the third half of today's show. That power of two again, so stay tuned. In the meantime, our number is 1-8-8-8-CAR-TALK. That's 8-8-8-2-2-7-8-2-5-5.
[ Car Talk Puzzler ]
