Weighing Designs to Detect a Single Counterfeit Coin

In this article, we discuss at length a combinatorial problem which has been of historic interest. It has appeared as a puzzle in several different versions with varying degrees of difficulty. It can be simply stated as follows: We are given a number of coins which are otherwise identical except that there may be at most one fake coin among them which is either slightly heavier or slightly lighter than the other genuine coins. Using only a two-pan weighing balance, we must devise a weighing scheme to identify the counterfeit coin and determine whether it is heavier or lighter (or declare that all coins are normal). We construct both sequential and non-sequential (that is, simultaneously declared) weighing plans for any given number of coins containing at most one fake coin using the minimum number of weighings needed.