This one jumped out at me. I know a guy who can solve the Rubik's Cube. Apparently he learned by going on Google.
I've seen a few "cubing" videos - there's a guy who solves ... actually, there are many Professor videos:
http://www.youtube.com/results?search_query=Rubik's+Professor
There are "codes" for how to solve it. It still takes 5+ minutes, but that's the slower, easier way. Apparently there are far more complicated "codes" where you can solve it in under a minute.
So that's what I'd suggest - do some quick Googling, and write down the easier codes. I'm pretty sure to solve it without help would require a massive amount of complicated math.
ETA: Maybe Kitty Pryde, our resident Mathemagician could help you with the math.
I solved it circa 1979-1980 (when this thing was first marketed and the Cube and its spinoffs were wildly popular for a short time) after owning it for six months, and reading some hints in the Mathematical Games column in Scientific American (there was a later cover article on the Cube). The main hint is to look for "macros." A macro is a sequence of moves, usually 5 to 7, that does something - often if you do it and then do its mirror image, often rotated by 90 degrees or a multiple, you will be left with a small number (like three) of changed cubes.
Once i found a method I practiced it, and could do it in three minutes regularly. My best time was 2:30, but there were teenagers (okay, I was an old guy in my early 20s) in TV contests who could do it in 30 seconds. I don't think they had as advanced algorithms back then as exist now, they just learned to do the similar macros at blinding speeds.
I remember how to do most of it, but there's one sequence I had of about 13 moves that I don't quite remember, and that keeps me from solving it nowadays.
I might suggest NOT looking up solutions online, but trying from what I wrote above, and see if you can solve it yourself without any more hints. If you do, it'll feel like a much bigger accomplishment.
The "mathematics" behind the Rubik's Cube is Group Theory, but I don't think it's of much help, unless perhaps you already know it very well (named the pieces in a circle starting with a corner piece: a, b, c, d, e, f, g, h and i, - when you rotate it, the corner pieces rotate - a goes to c, which goes to e, which goes to g, which then goes to the original a's position. The side pieces rotate the similarly. But this is obvious by observation...):
en.wikipedia.org/wiki/Group_theory
That's another thing that is/was kinda on my list, get a degree in mathematics (or more precisely, get the knowledge one would get while earning the degree), but my high school life was messed up to the point I was lucky to get into college at all, and it only offered engineering degrees. In the past ten years I read "Journey to Genius" which I think is a great book on mathematics and its history - if I had read it in high school it might have motivated me to want a math degree, but it wasn't even published until after I was out of college. I did want to do something in science, probably physics, but engineering was (almost) close enough.
s. Xmas is almost here. I'm getting excited 'bout it 
s Too funneh, Kyla! 
s
