A box is a set of the form X = X ₁×···×X _d , for some finite sets X _i , i = 1, . . .,d. Answering a question posed by Kearnes and Kiss [2], Alon, Bohman, Holzman and Kleitman proved [1] that any partition of X into nonempty sets of the form A ₁×···×A _d , with $ A_{i} \varsubsetneq X_{i} $ A_{i} \varsubsetneq X_{i} , must contain at least 2 ^d members. In this paper we explore properties of such partitions with minimum possible number of parts. In particular, we derive two characterizations of minimal partitions among all partitions of X into proper boxes. For instance, let P = P ₁×···×P _d be a fixed k-dimensional plane in X, that is P _i = X _i for exactly k different subscripts i, with |P _i | = 1 otherwise. It is shown that $ {\user1{F}} $ {\user1{F}} is a minimal partition of X if and only if P intersects exactly 2 ^k members of $ {\user1{F}} $ {\user1{F}} , for every such P.