# Discovery of higher dimensions rubik's cube



## charliemckiz (Apr 27, 2012)

First, I will list some rules that Rubik's Cube satisfies:
1. Rubik's Cube has 3 kinds of blocks, each has different number(1, 2 or 3) of color. 
2. The structure of each block reminds the same. 
3. 1-color blocks (or center) are relatively unchangeable. 
4.All cases of Rubik's cube satisfy:
i. The net rotation of all blocks is 0
ii. The permutation of all blocks is even.

By using analogy method and mathematical proof, I've got:
Theorem1: There are n kinds of blocks in a MCnD(Magic Cube n-dimensional). Each kind has different number of colors from 1 to n. 
Theorem2: The structure of each block reminds the same. 
Theorem3. 1-color blocks (or center) are relatively unchangeable. 
Theorem4: All cases of a MCnD satisfy:
i. The net rotation of all blocks is 0
ii. The permutation of all blocks is even.

Let's to further:
Let 1,2,..., n be the center of all the color of a n-color block. Since a self-spinning of a block is actually a permutation of 1,2,...,n, a spinning of a block can be written as: 
/ 1 2 ... \
\ f(1) f(2) ... / (Permutation notation)
We knew that single 3-color block spinning is illegal. 
In other words:
/ 1 2 3 \
\ 2 3 1 / and other equivalent cases are illegal in MC3D.
But this stands no more for MCnD. 
Modified Theorem4 i: When n≥3, single n-color block spinning is legal. 
When n=4, 
/ 1 2 3 4 \
\ 2 1 4 3 / and other equivalent cases are legal.
When n≥5,
/ 1 2 3 ... k ... n \
\ 2 3 1 ... k ... n / and other equivalent cases are legal.
Modified Theorem4 ii: When n≥4, permutation of k-color blocks are always even if k≥4.

By the way, I am the second one who have solved MC7D.
If you are interested in that, email me [email protected]. I will reply you as soon as possible.


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## Mike Hughey (Apr 27, 2012)

How does such a cube compare to the higher dimensional Rubik's cubes that already have been created in software?
http://www.superliminal.com/cube/cube.htm
http://www.gravitation3d.com/magiccube5d/


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## qqwref (Apr 27, 2012)

Why not just write the mathematical language here? We don't mind. (There is also a \( tag for LaTeX.) \)


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