For \(p\in N\) certain integer-valued functions \(A_ p(x)\), defined for \(x\in N\cup \{0\}\), are studied. These functions occur in a functional equation system corresponding to a generalized version of the transportation game ''Towers of Hanoi'' and their values may be interpreted as minimum numbers of moves. An explicit representation of \(A_ p(x)\) is given and so-called minimum partitions of x with respect to p are determined for all \(x\in N\). The minimum partitions of x are interest concerning the realisation of the minimum number of moves by optimal policies.