The original puzzleWhat standard can we apply to 4 weights which would enable us, with a simple 2-platter scale, to weigh all whole-number weights from 1 to 40. The AnalysisHere again, as for The Chain, we must build up the operation          from the smallest number of standard weights.     If we have only 1 standard weight, it must be of unit size, of weight "1",          and all it can do is measure an object of weight "1".     If we have 2 standard weights, the second must be of weight "3",           and we can measure objects of weights "3", and "4"               by placing one, or both, of our standard weights against the unknown.          We can measure an object of weight "2"               by placing the standard weight "3" against the unknown,                and the standard weight "1" with the unknown (acting as a subtraction).     If we have 3 standard weights, the third must be of weight "9",          and we can measure objects of weights from "5", to "13",          using the other 2 standard weights               in all possible combinations of addition and subtraction.     If we have 4 standard weights, the third must be of weight "27",          and we can measure objects of weights from "14", to "40",          using the other 3 standard weights               in all possible combinations of addition and subtraction. MathematicsThe choice of a new standard weight size is evidently made     by foreseeing the subtraction of the existing standard weights from the new standard :          when we have standard weight "1", we subtract it from "3" to weigh "2",          when we have standard weights "1" and "3",               we subtract them from "9" to weigh "5", and          when we have standard weights "1", "3", and "9",               we subtract them from "27" to weigh "14".If "n" represnts the number of standard weights,     the largest one will be of weight "3n-1".          and     the sum of the standard weights will be (3n - 1) / 2 .          With 10 standard weights, the largest would weigh "19, 683",               and the total weight would be "29, 524" ! PedagogyYou will find an application of this in the Devices Chapter.