Abstract
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.
Similar content being viewed by others
Suggested Reading
Freeman J Dyson, The problem of the pennies, Math. Gazette, Vol.30, pp.231–233, 1946.
Anany Levitin and Maria Levitin, Algorithmic Puzzles, Oxford University Press, 2011, Puzzle #10.
Mario Martelli and Gerald Gannon, Weighing coins: Divide and conquer to detect a counterfeit, The College Mathematics Journal, Vol.28, No.5, pp.365–367, 1997.
Robert L Ward, Finding one coin of 12 in 3 Steps, The Math Forum@Drexel: Ask Dr. Math, 1996, http://mathforum.org/library/ drmath/view/55618.html
Cedric A B Smith, The counterfeit coin problem, Math. Gazette, Vol.31, No.293, pp.31–39, 1947.
Richard K Guy and Richard J Nowakowski, Coin-weighing problems, Amer. Math. Monthly, Vol.102, pp.164–167, 1995.
Blanche Descartes, The twelve coin problem, Eureka, Vol.13, No.7, p.20, 1950.
Lorenz Halbeisen and Norbert Hungerbuhler, The general counterfeit coin problem, Discrete Math., Vol.147, pp.139–150, 1995.
P S S N V P Rao, Bikas K Sinha and S B Rao, Some combinatorial aspects of a counterfeit coin problem, Linear Algebra and its Applications, Special Issue, 2005.
Author information
Authors and Affiliations
Corresponding author
Additional information
Jyotirmoy Sarkar (left) is a Professor and Statistics Consultant at Indiana University-Purdue University, Indianapolis. His research includes applied probability, mathematical statistics and reliability theory.
Bikas K Sinha (right) retired as Professor of Statistics from Indian Statistical Institute, Kolkata. He has done theoretical and applied research in many topics in statistics.
Rights and permissions
About this article
Cite this article
Sarkar, J., Sinha, B.K. Weighing designs to detect a single counterfeit coin. Reson 21, 125–150 (2016). https://doi.org/10.1007/s12045-016-0306-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12045-016-0306-8