The Family Jewels?

Jul 04, 1998

RAY: Well here it is. Here it is. A struggling art history major waitress is working in New York City and it is around Christmas time and she finds herself in difficult straits. After all she's traveled home for Christmas and spent a lot of money and bought Christmas presents and the like, and she returns to her little apartment and her little room in the rooming house and realizes that she cannot pay the rent. So she approaches the landlord with the following, what would you call it, offer. She says, "Look. I don't have the money to pay you the month of January's rent." She said, "But I can give you a link of my gold necklace per day. Everyday I'll give you one link to pay for the rent." And as luck would have it there are 31 days in January and her gold necklace has 31 links.

TOM: Yeah.

RAY: You've got it.

TOM: Sure.

RAY: Now obviously he says, "Let me look at the thing." So he looks at it and verifies that it is authentic 18 carat gold, and he says you've got a deal. At the end of the month, I give you the necklace back, you give me the dough. Right.

TOM: Yeah.

RAY: Now obviously she could cut the thing into 31 pieces and every day, give him a piece, but then she'd have to have it repaired by a jeweler for an inordinate amount of money. So she wants to make as few cuts as possible. So what she's going to do is for example, day one she is going to cut one link. OK. She's going to hand him one link. Now she could cut another link off and hand him two for the next day.

TOM: Or she could just cut off two and give him the two and take one back.

RAY: There you go. And then for the third day, she gives him the previous one that she'd cut off.

TOM: She wouldn't have to cut anything.

RAY: So the question is how many pieces does she have to cut the chain into to be able to pay for all 31 days? How many pieces? So the chain is going to be -- now obviously when she makes the cut, she's going to close it back up again, you know. In other words, when she cuts that first link, she's actually cut link number two to extract that first piece. So that she doesn't lose it, she closes that back up again. Get the scenario.

TOM: I got it. Yeah. Yeah.

RAY: And then she's going to cut a two-link piece off and that's going to pay for day -

TOM: Sure. Sure.

RAY: Right now I've got it figured out for you - days one, two and three all figured out.

TOM: She did it with only two cuts instead of three cuts which would be the obvious straight-forward way to do it.

RAY: Right. Just cut a link at a time off.

TOM: Right.

RAY: So there's an economical way to do this. And the question is what are the lengths of the pieces that she has to cut the --

TOM: The fewest number of --

RAY: Obviously. The fewest number of pieces. And what are those various lengths. Now the hint is. The hint is that 20 years ago people would have had a much harder time solving this puzzle than they would today.

TOM: Excellent hint.

Answer: 
RAY: 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 the 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.

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