Using already known techniques along with some not so obvious innovations on my part, I was able to show (prove) that there are solutions for all K (except those of the form 9m+/-4 and 9m+/-5 which are impossible) for +/-K = +/- (x^3) +/- (y^3) +/- (z^3). A further stipulation is that x, y and z must be whole numbers that can be a combination of positives and negatives. This is achieved through simple subtraction. Setting up a table showing that all K can be represented using a multiple of 27 plus a mask lends validity to a portion of the proof. These representations may and often do contain many more than the required number of cubes summed up. I side step that problem by showing that no matter the K picked and how ever many cubes are required to create it in my representations, they can all be reduced to a maximum of cubes summed. Exactly what we require for the proof. Having done that we are complete. The three new cubes we have just reduced to are already included in table. They are items I have already represented in the above format.