We 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. Some results from [8] are improved by construction of a procedure which is proved to be optimal for all $n$'s belonging to the set \[ \bigcup_{k\geq2}([[3^kqrt6+1],4\cdot3^k]\cup[[3^kqrt2+1],20\cdot3^{k-2}]). \]