What is the popular challenge of solving the Rubik cube as quickly as possible using LEGO bricks?
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.