The Basic Construction of Addition and Multiplication Ghost Number Cubes

Addition Ghost Number Cube

You start out with what I call the six primary numbers that encircle the inner cube at all six faces. These primary numbers are the building blocks of the entire ghost number cube.

In the case of addition cubes, you simply add up to the corners, each pair of adjacent primary numbers.

Then, using the communative property where 1+3=3+1, for example, you add on both sides of each of the three pairs of primary numbers that fill out the three axis of the cube.

In the case of addition cubes, again, you add up to the corners of the outer layer cube.

The inner ghost numbers are colored purple. The outer invariant ghost numbers are colored red. The central ghost number is also colored red.

Circle the cubic array of numbers of both the inner ( purple) and outer (red) along the preferred spatial-diagonal to be studied.

Each summed inner ghost number pair ( colored purple) along a spatial-diagonal, in this case 12+30 =42: the outer ghost numbers ( colored red)

You can try the other inner ghost number pairs along the other three spatial-diagonals for yourself.

Multiplication Ghost Number Cube

Again, the six primary numbers are the backbone of the whole ghost number cube.

See if you can follow along, doing the same things as was done with addition cubes, but this time using multiplication, without too much further instruction.

(Hint: Look at each step in terms of development of the multiplication cube with nearly the same instructions for the addition cube.)

Each multiplied inner ghost number pair ( colored purple) along a spatial-diagonal, in this case 8 x 46,656 =373,248: the outer ghost numbers ( colored red)

You can try the other inner ghost number pairs along the other three spatial-diagonals for yourself.

There is a preferred angle of intersection among the inner ghost numbers.

They all add or multiply together to equal the outer ghost numbers, of course, but there is one preferred spatial-diagonal for addition cubes and two preferred spatial-diagonals for the multiplication cubes (colored yellow). The remaining inner ghost number spatial-diagonals ( colored brown for each cube) are copies that doesn't seem all that intriguing. I have yet to be fully satisfied with a reason for their appearances in the two types of ghost number cubes.

Here are a few examples of each: