weighings are sufficient
to find a bad coin among coins (Steinhaus 1999, p. 61).
vos Savant (1993) gives an algorithm for finding a bad ball among 12 balls in three
weighings (which, in addition, determines if the bad ball is heavier or lighter than
the other 11), and Steinhaus (1999, pp. 58-61) gives an algorithm for 13 balls.
Bachet's weights problem asks for the minimum number of weights (which can be placed in either pan of a two-arm balance) required to weigh any integral
number of pounds from 1 to 40 (Steinhaus 1999, p. 52). The solution is 1, 3,
9, and 27: 1, ,
3, , , , ,
, 9, , ,
, , , , , , and so on.
Bachet, C. G. Problem 5, Appendix in Problèmes plaisants et délectables, 2nd ed. p. 215, 1624.Ball,
W. W. R. and Coxeter, H. S. M. Mathematical
Recreations and Essays, 13th ed. New York: Dover, pp. 50-52, 1987.Bellman,
R. and Gluss, B. "On Various Versions of the Defective Coin Problem." Information
and Control4, 118-131, 1961.Descartes, B. Eureka,
No. 13, Oct. 1950.Dyson, F. J. "The Problem of the Pennies."
Math. Gaz.30, 231-234, 1946.Gardner, M. The
Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University
of Chicago Press, pp. 29-33 and 106-109, 1984.Kraitchik, M. Mathematical
Recreations. New York: W. W. Norton, pp. 52-55, 1942.O'Beirne,
T. H. Chs. 2 and 3 in Puzzles
and Paradoxes. Oxford, England: Oxford University Press, 1965.Pappas,
T. "Counterfeit Coin Puzzle." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 181,
1989.Smith, C. A. B. "The Counterfeit Coin Problem."
Math. Gaz.31, 31-39, 1947.Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, 1999.Strong, C. L.
"The Amateur Scientist: How to Make an Aerodynamic Smoke Tunnel and More about
the Puzzle of the 12 Balls." Sci. Amer.192, May 1955.Tartaglia.
Book 1, Ch. 16, §32 in Trattato de' numeri e misure, Vol. 2.
Venice, 1556.Tweedle, M. C. K. Math. Gaz.23,
278-282, 1938.vos Savant, M. The
World's Most Famous Math Problem. New York: St. Martin's Press, pp. 39-42,
1993.
weighings are sufficient
to find a bad coin among 
coins (Steinhaus 1999, p. 61).
vos Savant (1993) gives an algorithm for finding a bad ball among 12 balls in three
weighings (which, in addition, determines if the bad ball is heavier or lighter than
the other 11), and Steinhaus (1999, pp. 58-61) gives an algorithm for 13 balls.
,
3, 
, 
, 
, 
,
, 9, 
, 
,
, 
, 
, 
, 
, 
, and so on.
