The maximal number of regions into which space can be divided by n planes is f(n) = 1/6(n^3 + 5n + 6) (Yaglom and Yaglom 1987, pp. 102-106). For n = 1, 2, ..., these give the values 2, 4, 8, 15, 26, 42, ... (OEIS A000125), a sequence whose values are sometimes called the "cake numbers" due to their relation to the cake cutting problem. This is the same solution as for cylinder cutting.

cake cutting | cake number | circle division by lines | cube division by planes | cylinder cutting | plane division by circles | space division by spheres