Weighings

$n$ weighings are Sufficient to find a bad Coin among $(3^n-1)/2$ Coins. 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).

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. The solution is 1, 3, 9, and 27: 1, $2=-1+3$, 3, $4=1+3$, $5=-1-3+9$, $6=-3+9$, $7=1-3+9$, $8=-1+9$, 9, $10=1+9$, $11=-1+3+9$, $12=3+9$, $13=1+3+9$, $14=-1-3-9+27$, $15=-3-9+27$, $16=1-3-9+27$, $17=-1-9+27$, and so on.

See also Golomb Ruler, Perfect Difference Set, Three Jug Problem

References

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.

Kraitchik, M. Mathematical Recreations. New York: W. W. Norton, pp. 52-55, 1942.

Pappas, T. ``Counterfeit Coin Puzzle.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 181, 1989.

Tartaglia. Book 1, Ch. 16, §32 in Trattato de' numeri e misure, Vol. 2. Venice, 1556.

vos Savant, M. The World's Most Famous Math Problem. New York: St. Martin's Press, pp. 39-42, 1993.