There's no "race" as this has been solved multiple times by different people (solving rubik's cube isn't that hard). just take a look at the CubeStormer (or, better, the CubeStormer II).
I also saw building-plans and programs for such a cube-solver (not exactly the linked one). If you like to rebuild such a thing: I'll add links as soon as i find them.
EDIT:
As promised, there's the Tilted Twister with building plans online. See it in action on this youtube-video. This is very slow compared to the CubeStormer, but looks like it can be built using only the NXT2.0-pack and doesn't need thousands of parts and multiple expensive NXT-logic-blocks.
The plans for the tilted twister in your answer are awesome!
– jfyelle Oct 27 '11 at 13:45