The Amazing Math Inside the Rubik’s Cube | Science - Popular Mechanics

Dave Linkletter, Ph.D. candidate in Pure Mathematics at the University of Nevada, Las Vegas says, Want to solve the puzzle? Then you have to know the numbers.

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Next year marks 40 years of the Rubik’s Cube first hitting toy shelves. Ever since its release, it’s taunted almost a half billion tinkerers who think they can crack the confounding cube, only to be stymied by its maddening secrets. As we approach the Rubik’s Cube’s ruby anniversary (for real!) it’s time to unpack the puzzle once and for all—with deep math. Because while the cube’s literal insides may be made of plastic, its real guts are nothing but numbers. Let’s dive in.

Breaking Apart the Blocks

Starting with some basics, a 3x3x3 Rubik’s Cube has six faces, each a different color. The center of each face is attached to the core scaffold that holds the cube together, so they don’t move other than rotating in place. As a result, the same colors always end up opposite from each other; on a standard cube, white is opposite yellow, red opposite orange, and blue opposite green.

Bust open a Rubik’s Cube and you’ll see it’s made of three types of building blocks. First, there’s that central scaffold, connecting the center of each face. Then there are the cubies—the nickname for the little 1x1x1 blocks. The corner cubies have three colored sides, and the edge cubies have two. A Rubik’s Cube has one core, eight corner cubies, and 12 edge cubies.

The immediate math to be done with those numbers is the total number of ways you can scramble a Rubik’s cube: 43,252,003,274,489,856,000. Written in a more mathematical way, that number is (3⁸8!)(2¹²12!)/12. Here’s how that comes together...

God's Number and Beyond
The original challenge of the cube, of course, was solving it. Ernő Rubik made his first prototype in 1974, and early in the six years it took him to see it mass-produced, he was naturally the first person to ever solve the cube.

When the cube hit toy stores in 1980, some mathematicians had already been experimenting with early versions for a few years. One of them was Dr. David Singmaster, who wrote the famous guide “Notes on Rubik’s ‘Magic Cube’” and developed a writing method for describing turns of the cube’s faces. That notation has become the standard, and is now known as Singmaster notation. 
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Source: Popular Mechanics