Dependence of the number of positions each module can take on the number of modules m and the size of the problem z. The top line in each block shows all the positions that each module can occupy. Each subsequent line in the block shows a single valid position for each module (dark dashes, numbered according to module), along with the possible alternative positions for the last module (light dashes). This highlights the fact that the successive left-most positions of the last module correspond to successive sizes of the one-dimensional problem. Note that to keep the size constant it is necessary to add another spacer position for each additional module. The Current column shows the number of positions contributed by the current size of the (m − 1) dimensional case, while the Sum column shows the total number of positions for the current size of the m-dimensional case. Horizontal arrows show the contribution of each new term (larger size) to the sum: adding a base of spacer is the same as adding the case with the new number of bases of spacer in one lower dimension. Horizontal and vertical arrows show that each successively larger term in a given dimension is the sum of the previous term in that dimension and the larger term in one fewer dimension. Oblique arrows show the relationships between terms in successive dimensions.
