Searching for two counterfeit coins with two-arms balance

Abstract

The following search model of a coin-weighing problem is considered: G has n coins, n - 2 of which are good coins of the same weight w g , one counterfeit coin of weight w h is heavier and another counterfeit coin of weight w l is lighter ( w h + w l = 2 w g ). The weighing device is a two-arms balance. Let N A ( k ) be the number of coins for which k weighings suffice to identify the two counterfeit coins by algorithm A and let U ( k ) = max { n ∣ n ( n - 1 ) ⩽ 3 k } be the information-theoretic upper bound of the number of coins; then, N A ( k ) ⩽ U ( k ) . One is concerned with the question whether there is an algorithm A such that N A ( k ) = U ( k ) for all integers k. It is proved that the information-theoretic upper bound U ( k ) is always achievable for all even integer k ⩾ 4 . For odd integer k ⩾ 3 , we establish a general method of approximating the information-theoretic upper bound arbitrarily. The ideas and techniques of this paper can be employed easily to settle other models of two counterfeit coins.