Jul 13, 2013
RAY: 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.
One day, a neighbor borrows the stone. He had to apologize when he returned it because it had 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.
The question is, 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?
And here's the hint: how would you weigh two pounds?
RAY: Here’s the answer. 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. 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. And now we're in a lot of trouble.
RAY: Well, you're not. And the only way I came to the answer is 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. One, three, nine and twenty-seven.
TOM: I like it!