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Ever since the Rubik’s Cube was released, it’s taunted almost a half billion tinkerers who think they can crack its confounding mysteries, only to be stymied by its maddening secrets. Now, it’s time to unpack the puzzle once and for all—using some deep math. The cube’s literal insides are made of plastic, but 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.
The first term, 38, counts every way the eight-corner cubies can be rotated. A corner cubie can fit into its slot, rotated three different ways. That’s a factor of 3 for each of the eight corner cubies, so they multiply to 38.
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Photo by Andrew Spencer/Getty Images
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