Mar 13, 2021
RAY: This was sent in by Dave Etnoyer.
A farmer had a 40-pound stone which he would use to weigh 40 pounds of feed. He had a balance scale; he would put the stone on one side and pile the other side with feed, and when it balanced, that was it.
A neighbor borrowed the stone, but 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 had actually 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.
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: Clearly, one of the pieces has to be one pound. I think we all agree on that.
And I think you guys thought was the next one will be three. Put one on one side and three on the other and that's two. Three pounds is obvious because he's got the three-pound thing. Four pounds is easy: he puts the three and the one together. And now we're in a lot of trouble. I'm going to say one, three, five, and whatever's left, 29 or something.
The way I stumbled upon the answer was that somehow I figured out it had to be powers of three. 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. And that's what they are. One, three, nine, and 27. I think those are the only ones that work.
