MikeInNH — "No...it's probably because cartalk is the only thing google can find on the subject."Not true. The Google quest must be properly phrased to return a solution.
MikeInNH — "Here's a hint....Hopefully it won't give it away."That is not a hint at all, because one would still be left in choosing, presumably at random, four weights that could be placed on a balance scale to balance any integer weight between 1 and 40 lbs.
Warning! Hint given below.
Proceed more methodically. Find the smallest stone, of weight W(2) lbs, that can be broken into two parts, such that the two smaller broken pieces can balance any weight from 1 to W(2) lbs. Then proceed to find the next smallest weight, W(3) lbs, that can be broken in to three pieces, etc. … and proceed in that fashion to find the smallest weight, W(n) lbs, that can be broken into n pieces that can be used to balance any weight from 1 to W(n) lbs. (Obviously, W(4) = 40 lbs.) By solving initially the n=2 problem, a general solution will emerge.