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I've been interested in big cubes (6x6x6 and larger) for a while now and my favorite so far is the 20x20x20. Some basic facts on the 20x20x20 cube, it's an even cube and follows the rules of all other even cubes. The number of combinations to a 20x20x20 cube is absolutely enormous. Below is an exact calculation of the number of possible combinations to a 20x20x20 cube.

This works out to about 1.3366106203729717328004388722211 * 10^1477 possible combinations. Click here for an idea of how large that number is. An interesting thing to note is that out of all the possible combinations on a 20x20x20 there is more than one "solved" state. This is due to the fact that if the centers are solved the pieces on that face can be in many different combinations and the face still appears solved. Below is the exact calculation of the number of solved states,

That's about 2.5074588992487256829778381899096 * 10^646 solved states! So for the 20x20x20 cube there is only one possible state where every face is a solid color, however practically when solving the cube you can piece that state together in the above number of ways. This is similar to having the center pieces on a 3x3x3 rotated while the cube is solved. The cube will still be solved with ever face a solid color, however the centers can rotate into 2048 different states without you ever knowing they were rotated! If you're interested in the number of combinations to different sizes of cubes where the position of every center piece is noticeable, called the n x n x n supercube, then check out my formula page for some formulas I came up for the number of combinations to different types of cube related puzzles. A supercube would be a cube that has a picture on every face, with the pictures designed like a jigsaw puzzle. In order to solve the cube, every center piece on the entire cube must be returned to its original position. This leaves exactly 1 solution out of many more possible combinations than the normal n x n x n cube.

I recently solved it and took screen captures throughout my solution and here you can watch how exactly a 20x20x20 cube is solved, or at least one possible way (there are many ways to solve rubik-like puzzles). My solution to the 20x20x20, and in fact all large cubes is almost identical to my 4x4x4 solution. Step 1 is to solve all of the centers, step 2 is to solve all of the edges, and step 3 is to solve the cube as if it were a rubik's cube (3x3x3). I've found that this method works very well for large cubes as the centers are the hardest part to solve, so I start with them while I still have the most mobility on the cube. The edges are easy to solve once all of the centers are solved, using the same principle as in my 4x4x4 solution. Finally the end solves like a rubik's cube except for having to correct the parity of the centers and edges - this is commonly reffered to as the "parity error". These errors are due to the fact that the parity of each center and edge orbit can vary depenging on the type of moves you do. I'll soon post a mathematically rigorous solution to the NxNxN cube and in it I will have a detailed discussion of the parity of the centers and edges and how that causes these "parity errors". Below starts the slide-show of the screen captures for my solution. If you would like to see the full size picture (1024 x 768) then click on the thumbnail. Enjoy!

So if you're ready click on start to see how its done

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