How to calculate / describe relative position (rubix cube) - algorithm

Question

This is an algorithmic problem. I can't seem to find a way to compare relative positions of 2 cubes in a rubix cube.
I've numbered all the 20 cubes in my program. and I'm using their this coordinate system, but now that I wanted to model two cubes in relative position I'm having trouble.
For example, say I saw the two cubes I'm watching in position 8 and 10, then later I saw them in position 12 and 13, well in both situations they're both on the same face of the cube, and they're both across from each other, not adjacent. Relatively speaking, that's the same representation of their location.
(By the way I'm only concerned with the "edge cubes" at this point, that's not the corners, so: 8 10 9 11 12 13 14 15 16 17 18 19 positions).
So anyway I thought if I listed every position in relation to each staring point, using the same algorithm to list each one, then I could compare the indexes and if they were the same, the relative position would be the same (but I was wrong, I might be on the right track, but it doesn't always work):
08 10 18 16 12 13 14 15 09 11 19 17
09 11 19 17 13 14 15 12 10 08 16 18
10 18 16 08 14 15 12 13 11 09 17 19
11 19 17 09 15 12 13 14 08 10 18 16
12 13 14 15 11 19 17 09 16 08 10 18
13 14 15 12 08 16 18 10 17 09 11 19
14 15 12 13 09 17 19 11 18 10 08 16
15 12 13 14 10 18 16 08 19 11 09 17
16 08 10 18 19 17 09 11 13 12 15 14
17 09 11 19 16 18 10 08 14 13 12 15
18 16 08 10 17 19 11 09 15 14 13 12
19 17 09 11 18 16 08 10 12 15 14 13
Consider the following two positions: cube A is at potion 19 and cube b is at 16. they're adjacent on the bottom level. Here's "19" row and it's indices to 16:
0 1 2 3 4 5
19 17 09 11 18 16 08 10 12 15 14 13
Now compare that to the relative position of the cube c and d at 13 and 9. C and D are adjacent on the right side, so they should have the same relative position. But my method doesn't determine that.
0 1 2 3 4 5 6 7 8 9
13 14 15 12 08 16 18 10 17 09 11 19
index 6 is not equal to index 9. Anyway that was my best approach and it took all day to come up with.
Does anyone have any other strategies that come to mind for calculating / expressing relative position between two locations on a cube?
Thanks very much for your help, and consideration on this topic!

Answer 1

There are two problems here:
I think you made a mistake when you calculated the relative positions from cube 13. I get:
0 1 2 3 4 5 6 7 8 9 10 11
13 14 15 12 17 09 11 19 08 16 18 10
This lines up with the other one, so cube 9 occurs at position 5. Compare this with the first row:
0 1 2 3 4 5 6
19 17 09 11 18 16 08 10 12 15 14 13
As required, cube 16 also occurs at position 5 (I think you mixed something up in your question. You mention index 6 when you mean 5. You number the indexes up to 6, but at position 6 there is cube 8, not cube 16. Please check that again).
The second problem is that given only a cube position without a reference cube for the orientation, there are two ways to number the cubes. Since your cube is not colored, you can rotate the cube by 180 degrees and come to another numbering for the reference cubes. Given that the relative positions for cube 19 are correct, I can also number the relative positions for cube 13 like this:
0 1 2 3 4 5 6 7 8 9 10 11
13 12 15 14 08 16 18 10 17 09 11 19
Note that this is close to your version but indexes 1 to 3 are in a different order. I think you were not consistent in the way you looked at the cube.
The main problem already becomes apparent in this paragraph:
For example, say I saw the two cubes I'm watching in position 8 and
10, then later I saw them in position 12 and 13, well in both
situations they're both on the same face of the cube, and they're both
across from each other, not adjacent. Relatively speaking, that's the
same representation of their location.
For every cube, there are two other cubes being on the same face and across from each other. To eliminate this ambiguity, you have to take orientations into account or reduce the number of relative positions (e.g. index 1 and 3 in your current scheme would denote the same relative position).