Once again you succeed to confuse me: THEN you can NOT begin with any number but with a cube, since you want

that this number, when cubed (once again) , we'll get:

(1) a cube (again!),

(2) the very same number we began with! (Of course, only condition (2) is required here since

condition (1) follows at once if we begin with a cube, but I'm already wary of all this so I'd better

be repetitive...)

In the body of your explanation you wrote "Now my challenge is to take a number, cube it, and then

add up its digits to see if you get ==>> another cube number <<== from the digit summands" (the

remarked part is mine), but then you say, in your PS, that that sum must be THE same number you began with!

Thus, your example 43 was NOT a good example (nor bad one: not a example at all but a misleading

piece of data) since 43 is not a cube.

Examples (of what you, hopefully, want) would be 0, 1, 8, 27, ...right?

Tonio