The array of numbers which represent the treatment combinations of factorial arrangements are utilized to construct balanced incomplete block designs and partially balanced incomplete block designs with two associate classes. A systematic procedure of selecting portions of these arrays is presented. These partial arrays lead to the construction of various incomplete block designs when the columns of the array denote the blocks (treatments), the rows of the array denote the treatments (blocks), and the presence of a specific number in a particular column (row) denotes the presence of a treatment in the block which that column (row) represents.