Searching for two counterfeit coins with two-arms balance

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 wg, one counterfeit coin of weight wh is heavier and another counterfeit coin of weight wl is lighter (wh+wl=2wg). The weighing device is a two-arms balance. Let NA(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)⩽3k} be the information-theoretic upper bound of the number of coins; then, NA(k)⩽U(k). One is concerned with the question whether there is an algorithm A such that NA(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.