Counterfeit Coin Problems

Since such weighing problems are today as much a part of the tradition of recreational mathematics as magic squares and mobius bands, it is interesting to note that they date only from that problem in 1945. The classic works of Loyd, Ball, Dudeney and Kraitchik contain no such problems. The responses to Schell's problem, a flurry of papers in the Monthly, Scripta Mathematica, and the Mathematical Gazette, contain no mention of earlier publications which might be relevant. Thus, it is apparent that this class of extremely natural and appealing puzzles is a recent invention, not an old chestnut which "crops up from time to time to puzzle and infuriate new generations of solvers" as someone wrote in 1961 (when such puzzles were just fifteen years old!). That early spate of papers, appearing in 1945 and the next few years with a speed unheard of in these days of publication backlogs, solved, resolved, and generalized the original problem in all directions. In this paper I present several variations of the balancing problem, all of which have been solved before. The method of solution given here may be original. We will always be dealing with a set of coins, of identical appearance, and a beam balance. We are interested in minimizing the maximum number of weighings which may be required to find the odd coin. The method of solution chosen may require solution of several types of weighing problems simultaneously, because a problem may change character after a weighing. For example, after a single use of the beam we have some coins which we know to be genuine (those on the beam, if it balances, those left off it if it does not). Thus, after one weighing we have a problem of a different type than the original one. We shall break all of the problems dealt with into two classes: those in which the counterfeit coin is known to be underweight and those in which it is only known to be of a different weight than the genuine coins. At the very end of the paper we will examine the possibility of no counterfeit coin. For the time being, we will assume that in every case exactly one coin is not genuine. Sometimes a "standard" coin, known to be the correct weight, will be provided. We begin with the first class of problems.