From the weekly car talk puzzler:

RAY: 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.

…

RAY: 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.

…

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?

TOM: Oh…

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.

Think you know? Drop Ray a note!

Oooh! I was leading a problem solving thingie with some teachers at my school, and I chose this problem to see if I could develop a way for the teachers (science, math) to get to the solution without telling them.

http://samjshah.com/2010/05/03/weights-goldsmiths-optimization/

It was a really fun day, but man was I worried their frustration was going to lead to them giving up!

Sam

That’s great Sam!

I do find these type of problems frustrating sometimes, it can feel like you’re getting nowhere and then all of a sudden you have the answer. 0 to hero.