Calculate Integer Arrangement

Name: Calculate Integer Arrangement
Author: Jürgen Kummer

Calculator for the different possibilities to arrange a certain amount of whole pieces evenly in length, width and height. Also the number of different arrangements will be determined.
Please select first, if the arrangement should be two- or three-dimensional. Then enter the number of pieces. The arrangement calculated last, at the very bottom of the list, usually is the most compact one. At a large amount and three-dimensional arrangement, the calculation might take a while.

Example: one possible way to arrange 12 pieces two-dimensional is 3 x 4. One possible way to arrange 24 pieces three-dimensional is 2 x 3 x 4. The slanted arrangement for the three-dimensional case is only for perspective purposes, to make the third dimension visible.

Here it is assumed that each object occupies the same amount of space. The size and shape of the objects themselves are irrelevant. The only requirement is that each object is located either exactly next to another object in each of the six directions, or next to no object in one, two, or three directions. In the latter case, it is a border object. If it has no neighbor in one direction, it is located on a side, but not at its boundary. If it has no neighbor in two directions, it is an edge object, and if it has no neighbor in three directions, it is a corner object.
The number of arrangements does not differentiate between the individual dimensions. In the arrangement 2 x 3 x 4, for example, the 2 can be the length, width, or height. The same applies to the other two values. Therefore, 2 x 3 x 4 means exactly the same as, for example, 3 x 4 x 2. The sorting is always done by size, from the smallest to the largest number.