Christopher Mowla Math 3900 November 18, 2011. The Rubik’s Cube. Rubik’s Cubes of Order n. A Rubik’s Cube of order n is referred to as the n x n x n cube, where A cube with n = an even integer value is referred to as an even cube .
November 18, 2011
As with the mathematical definition of permutations, the term permutation is also used to describe the location where each piece is located.
The first row of the permutation notation can be assigned to fixed slots on a cube.For example, the slots containing the wing edges in the last layer of a 5x5x5 can correspond to the permutation notation as follows:A second meaning of the term “permutation” on Rubik’s Cubes
Each number on the cube designates the slot number (the first row of the 2-line permutation notation) and f(1) is the wing edge piece in slot 1, f(2) is the wing edge piece in slot 2, etc.
Orange one corner out of the 8 to be in the identity, then the sum of the values of either +90 degree twists or -90 degree twists (not both) from the identity must be evenly divisible by 3.
For odd cubes, notice that the white cross also consists of 4 symmetrically equivalent blocks. If we rotate the face 3 times, all of these
blocks will go to each other.
(These blocks make up the orbits of the + center pieces.)
Each face is a different color.
Non-fixed center pieces are indistinguishable.
Therefore, we must divide by:
This gives us a final number of:Non-Fixed Centers Final
Notice that the numbers of the center pieces are all turned to match the direction of the numbers on the corners and middle edges perfectly.Supercubes
There are 6 fixed centers on every odd cube. sizes begin with
Each center can be rotated in 4 directions in its location.
This gives total permutations.
By the “cube law,” we divide this number by 2, because only half of the fixed center permutations are allowed:Fixed Centers (Odd Cubes Only)
From calculating the formula for the regular sizes begin with nxnxn, we found that there are:orbits of non-fixed center pieces.
In each orbit, there were 24/6 = 4 center pieces of each color.
Now that center pieces ARE distinguishable, we have an additional 4! permutations of each color. There are 6 colors, which gives usNon-Fixed Center Pieces
Setting the first derivative equal to zero to find the critical point, we have:
Substituting w into , we get
By comparing the formula to the true maximum values, we just need to take the floor of this.
We can see a 3-D graph representation of the formula critical point, we have:
very well by rewriting as .
w critical point, we have: