RAY: Hi. We're back. You're listening to Car Talk with us, Click and Clack, the Tappet brothers, and here is the answer to last week's puzzler. Yeah. Did that dope slap I gave you jog your memory?

TOM: Not a thing.

RAY: Did it injure your brain cell? After all, it's sloshing around in there.

TOM: Tell the truth. Tell the truth. Do you actually remember the puzzler every week, or do you have to look at your little notes there?

RAY: Yes. Nope.

TOM: I see you got little notes.

RAY: I refer to my notes so that I don't mis-state anything. So that the re-statement of the puzzler is exactly the same --

TOM: But when you walked into the studio, if I had said to you what was last week's puzzler you'd have it, huh?

RAY: There you go.

TOM: I'm going to try that next time. If we ever do another show, I'm going to ask you.

RAY: OK. Here it is. A struggling young art history major/waitress is working in New York City around Christmas time, and she finds herself in difficult straights. After all, she's just traveled home, bought presents and the whole bit, and she returns to her little room in the rooming house and realizes that she can't pay the rent for the month of January. So she approaches the landlord with the following proposition. She says, look, I don't have any money to pay you the month of January's rent.

TOM: Oh, the bracelet one.

RAY: Very good.

TOM: It took me a long time.

RAY: You were close. Bracelet for the neck.

TOM: A bracelet for the neck, yeah.

RAY: Anyway, she says, but I can give you a link of my gold necklace for every day that the rent is late. As luck would have it there are thirty one days in January and her necklace has what --

TOM: 31 links.

RAY: 28 links. 31 links. And he says, let me look at the thing, will yah. He looks at it and verifies that it is authentic 18 karat gold and he says you've got a deal. At the end of the month when you give me the dough, I'll give you back all the pieces. Now obviously she could cut the necklace into 31 pieces and every day she would give him one piece, but then at the end of the month she would have to have the necklace repaired by a jeweler which would cost probably a month's rent. So she wants to make as few cuts as possible to this necklace, so what she is going to do is day one she is going to cut off one link and give it to the guy. Now she could cut another link off and give him the second link on the second day, but she decides to cut off two and give him that pair of links. See.

TOM: Sure. And he is going to give her back that she gave him yesterday.

RAY: Take one back.

TOM: So he's going to have two.

RAY: Then the third day, she gives him --

TOM: She doesn't have to do any cuts, she gives him the one back. He keeps the two and the one; he's got three.

RAY: There you go. OK. Now you got the idea.

TOM: I got the idea.

RAY: So she doesn't have to cut anything off for the third day. I mean she's got two and one and you've got that figured out. OK. So the question is very simply, what is the fewest number of pieces that she can cut the chain into in order to pay for thirty one days. And what are the lengths of those various pieces. Now, we already know she needs one for day one. And she needs two. Well, it turns out that every integer could be expressed as the sum of powers of two. Right.

TOM: Yeah.

RAY: Right. And between one and thirty-one how many powers of two are there?

TOM: There are four, plus the fifth one.

RAY: Well, there's two to the zero which is one. There's two to the first which two. There's two to the second which is four. There's two to the third which is eight. And there's two to the fourth which is 16, and then two to the fifth is 32, but she doesn't need 32 pieces.

TOM: That's what I just said. There's four plus the fifth one which would be 32.

RAY: So how many pieces does the chain have to be cut in?

TOM: Four. Five.

RAY: Five. Four is close enough. If you're not really paying attention, four's good.

TOM: Yeah, I mean she might come into money around the 18th of month.

RAY: You never know. And the hint I gave was that twenty years ago people would have had a more difficult time solving this, because people were not binary literate as much as they are today. I mean everyone knows.

TOM: I still don't get the binary thing. What's that got to do with it.

RAY: It's all powers of two. You know, base.

TOM: Yeah. So she's going to have one.

RAY: Oh, I knew this was going to be a problem.

TOM: Two, four, eight and sixteen.

RAY: 16. And you can make every number between 1 and 31 with those five pieces.

TOM: Of course you can, can't you.

RAY: Yes you can.

TOM: Isn't that good.

RAY: OK. Who's our winner this week?

TOM: The winner this week is Suzanne Boening.

RAY: Oh, they make great planes up there at Boenings, don't they.

TOM: Boning, yeah. From Tuscaloosa, Alabama.

RAY: Wow, no kidding.

[ Car Talk Puzzler ]