# COMPOSITE CUBES

It should be possible to make pan-magic cubes from base lines for both even and odd order where the order is a composite number based on concepts developed for the comparable squares. Order-12 nasik cubes are known. (See Nakamura's Site) Odd orders singly divisible by 3 will probably not be possible just as comparable figures were not possible for squares. This may include odd orders doubly divisible by 3 for cubes. (I have been unable to make an order-9 nasik cube using base lines.) This area needs further exploration.

## ORDER-27 PAN-MAGIC CUBES

Magic cubes can be built using ternary base lines, provided they are of order-3^{p} where p ≥ n. The master base lines for two order-27 pan-magic cubes are given in the 27 cube worksheet of the *CubeLines* Excel Spreadsheet available on the Downloads page. The cubes themselves are too large to be included on that worksheet. The Excel converter to a cube and associated checker are available by request to the author.

Both cubes are compact by that terms definition for cubes made using ternary base lines, i.e. the sum of every number in every 3x3x3 sub-cube within the larger cube adds to the magic constant. Also evenly spaced grids of 27 numbers in 5x5x5, 7x7x7, etc sub-cubes will also add to the magic constant although this is not shown in the checker. Rectangular prisms with the correct spacing also add to the constant. Both cubes are also associated. Cube 2 was made by using a modification of the structured magic figures discussed in n-Dimensions.