I found both the outer and inner ghost numbers this year after 26 years since 1994 in college algebra class where I invented them!!!

I readily blame it on pure laziness. I knew there had to be outer ghost numbers but I thought the calculations would be too hard to discover if I was right and I thought a calculator would be cheating, even for the multiplication cubes where the numbers grow exponentially.

What I didn't realize was that there was also a hidden inner ghost number that completed the ghost cubes and their addition/multiplication 1/3 and cubed inverse relation to the outer ghost number proved that they were the correct way to determine the inner ghost numbers.

The inner ghost numbers are the gold cubes inside the inner ghost cube with those ghost numbers written on them in barely discernible silver pen.

I'm waiting for my 300 smooth styrofoam balls to come in from Amazon on the 22nd to build my inner ghost number cubes.

Ghost Cube with emphasis on the inner ghost number. In this case, 14.

27 foam balls needed per cube, and there are six cubes to a set of addition/multiplication 1234, 2468 and 3579 ghost cubes to render in inner ghost number form. 162 foam balls in all need for all six cubes. This is an example.

It is my belief that I have indeed isolated an inner ghost number, based on their 1/3 for addition and cubed for multiplication relationships to their outer ghost numbers.

It is also my belief, that with all 27 numbers of the cube accounted for, you can discard the inner ghost cube and have a Magic Ghost Cube in some sort of matrix form that may be useful, once the primary numbers around the cube is found to have some meaning for application to some real world problems. For now, I just semi-randomly place six numbers around these cubes to study, usually as 1234, 2468 and 3579 addition and multiplication cubes.

And since all Faces, Edges and Vertex ghost numbers are all accounted for in these Magic Ghost Cubes, with the inner ghost number to boot ( not shown in the diagram below) , they may have an answer to their origin in a study of topology.

Keith Jason Carlock, Inventor of the Ghost Number Cube?? ( so far, without an application in the real world ?? )