We consider the problem of ascertaining the minimum number of weighings which suffice to determine all counterfeit (heavier) coins in a set of $n$ coins of the same appearance, given a balance scale and the information that there are exactly four heavier coins present. A procedure which is either optimal or suboptimal is constructed for an infinite set of $n$'s. For another infinite set of $n$'s a procedure is constructed for which the maximum number of steps differs by just two from the information-theoretical lower bound. We also consider a slightly modified problem, i.e. the case when we are given a certain number (not greater than $n$) of additional coins for which we know that they are all good (not counterfeit). For that case, and arbitrary $n$, we determine an upper bound for the maximum number of steps of an optimal procedure which differs by just four from the information-theoretical lower bound. The proofs are given by an effective construction of a procedure.