Puzzle: The 40 pound stone

This is a variation of a puzzle given on Car Talk. It’s similar to some of the puzzles in my book Interview Puzzles Dissected.

A farmer uses a double-pan balance and a 40 pound stone to measure 40 pound weights: grain, flour, turnips, whatever. One day he loans his weighing stone to a neighbor who accidentally breaks it into four not-necessarily-equal-sized pieces.

After sulking for a few days, he tells his neighbor, “I have to thank you. With the four pieces I can now weigh any integer weight between 1 and 40 pounds!”

What are the weights of the pieces?

This puzzle is similar to the pan balance puzzles described in my book, Interview Puzzles Dissected. It explains how these puzzles work, describes variations on the puzzles, and tells you what the “trick” is for solving other similar puzzles.

The book’s sample puzzles page describes and solves three different kinds of puzzles.

(I’ll post the solution to this puzzle in a day or two.)