Discovery of higher dimensions rubik's cube
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 charliemckiz@rocketmail.com. I will reply you as soon as possible.
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