Abstract. For integers $m_1,\ldots,m_d>0$ and a cuboid $M=[0,m_1]\times\cdots \times[0,m_d]\subset{\msbm R}^d$, a set $H$ of closed bricks in $M$ is a system of brick islands if, for each pair of bricks in $H$, one contains the other or they are disjoint. Such a system is maximal if it cannot be extended to a larger system of brick islands. We show that the minimum size of a maximal system of brick islands in $M$ is $\sum_{i=1}^d m_i - (d-1)$. Also, a system of cubic islands is a system of brick islands for which all the bricks are cubes. We show that the minimum size of a maximal system of cubic islands in a cube $C=[m]^d$ is $m$.

AMS Subject Classification (1991): 05A05

Keyword(s): brick islands

Received December 17, 2010, and in revised form April 19, 2012. (Registered under 89/2010.)