Oct 22, 2011
RAY: We're back; you're listening to Car Talk with us, Click and Clack, the Tappet Brothers, and we're here to discuss cars, car repair and the new puzzler. Now, I have several potential new puzzlers.
TOM: Oh, you haven't actually decided yet -- you're going to do it right here, live, on National Public Radio? You're going to a take out the two or three that are in contention, so to speak, and you're going to just decide right now? And what are we supposed to do in the meantime? We're supposed to just sit here and stick our fingers in our ears?
RAY: Yeah, like you usually do. Well, I tried a couple on the staff and they didn't get any of them, so I'm going to try this one, which I didn't try on them, because I like this one the best. This is a non-automotive puzzler. This was sent from someplace on Alpha Centauri by Dave Etnoyer. Here it is.
A farmer had a 40-pound stone which he could use to weigh 40 pounds of feed; he would sell feed in 40 pounds, or bales of hay, or whatever. He had a balance scale; he put the stone on one side and pile the other side with feed or hay, and when it balanced, that's it.
TOM: Hmm, another one of these.
RAY: Well this is interesting.
TOM: I like it already.
RAY: Don't start with the liking it! I don't want to hear you like it -- I'm sick of that.
TOM: I don't want to build up your hopes, but I do like it so far. The fact that it's a stone, and it weighs exactly 40 pounds; I find that interesting.
RAY: Well, they chip off pieces!
TOM: -- make the stone. You go into Brookstone and say, "I want a 40-pound stone."
RAY: Who else do you buy it from? Brook-STONE, right.
TOM: Do you deliver?
RAY: You go into the brook, you pick out a stone --
TOM: I just want to tick off the UPS guy when they have to deliver it --
RAY: Can I continue?
TOM: I'm sorry, did I interrupt?
RAY: Then you screw up the puzzler -- you interfere with the very nature of it, and you have the audacity the next week --
TOM: No, this is serious. The puzzler is the only serious part of the show. OK.
RAY: OK. A neighbor borrows the stone, but he had to apologize when he returned it, broken into four pieces. The farmer who owned the stone later told the neighbor that he actually had done him a favor. The pieces of the broken stone could now be used to weigh any item, assuming those items were in one-pound increments, from one pound to 40.
TOM: Or 41? How about that! Now that would be a puzzler.
RAY: Yeah, that would be good. What were the weights of the four individual stones? So if you want to weigh one pound, six pounds, 11 pounds, 22 pounds, 39 pounds -- how would you use the stones, the thing you are weighing, and the balance beam?
RAY: Remember that. And here's the hint: how would you weigh two pounds? That's the question. I could give a further hint --
TOM: No, don't. That is great!
RAY: Yeah, till next week. Next week it'll be in the dog house.
RAY: Clearly, one of the pieces has to be one pound. I think we all agree on that.
TOM: And I think the next one will be three.
RAY: That's good. Why do you think so?
TOM: Put one on one side and three on the other and that's two.
RAY: Huh? You put the one pound weight on one side to weigh a two pound piece, the two pound thing on that same side and the three pound weight on the other side.
TOM: Exactly. Three pounds is obvious now because he's got the three pound thing. Four pounds is easy, he puts the two together --
RAY: He's got three in one and --
TOM: And now we're in a lot of trouble.
RAY: Well, you're not. And the only way I came to the answer --
TOM: Well, I'm going to go one, three, five and whatever's left, 29 or something.
RAY: Well, that's close but it's wrong.
TOM: 31. No we don't need five because we can play around to get five with a bigger number. We could have an eight and we could do five --
RAY: Ooh, you're so close!
TOM: And we could have a six and have --
RAY: Well, the way I stumbled upon the answer by figuring there has to be --
TOM: By tripping over the book that the answer is in!
RAY: That's what it was, it was in the Math group. That somehow I figured out it had to be powers of three, because if it broke into four pieces, there are four powers of three between one and 40 -- three to the zero which is one, three to the one which is three, three squared which is nine and three cubed which is 27.
TOM: They don't add up to 40 by any chance?
RAY: They do!
TOM: Oh, my God! So they do.
RAY: And that's what they are.
RAY: One, three, nine and 27.
TOM: Or nine if you put four on one side and nine on the other, that gives you the five.
RAY: Trust me it works. It works. I don't think they're are any other four sides that will allow you to do this, but I know one, three, nine and 27 do and I think those are the only ones that work.
TOM: This is like the Lou Gehrig thing.
RAY: Well, actually this is similar to the -- similar to but not quite like the puzzler that we had about the necklace some time ago. Taking pieces of the necklace --
TOM: Ah, yes, it's very similar.
RAY: It's very similar. Requires the same kind of thought process, except this required turning to page 18 where the answer was. And who's our winner this week, Tommy?
TOM: Oh, we got a winner?
TOM: Let me look on this little piece of paper. The winner is John Hengesbach from Windham, New Hampshire.