Two counterfeit coins

AbstractWe consider the problem of ascertaining the minimum number of weighings which suffice to determine the counterfeit (heavier) coins in a set of n coins of the same appearance, given a balance scale and the information that there are exactly two heavier coins present. An optimal procedure is constructed for infinitely many n's, and for all other n's a lower bound and an upper bound for the maximum number of steps of an optimal precedure are determined which differ by just one unit. Some results of Cairns are improved, and his conjecture at the end of [3] is proved in a slightly modified form

Optimal detection of two counterfeit coins with two-arms balance

AbstractWe consider the following coin-weighing problem: suppose among the given n coins there are two counterfeit coins, which are either heavier or lighter than other n−2 good coins, this is not known beforehand. The weighing device is a two-arms balance. Let NA(k) be the number of coins from which k weighings suffice to identify the two counterfeit coins by algorithm A and U(k)=max{n|n(n−1)⩽3k} be the information-theoretic upper bound of the number of coins then NA(k)⩽U(k). We establish a new method of reducing the above original problem to another identity problem of more simple configurations. It is proved that the information-theoretic upper bound U(k) are always achievable for all even integer k⩾1. For odd integer k⩾1, our general results can be used to approximate arbitrarily the information-theoretic upper bound. The ideas and techniques of this paper can be easily employed to settle other models of two counterfeit coins