Above is a simulation for a Rubik's cube with 10 dimensions after shuffling 100 times generated in this project.
Rubik cube is 3D pazzel invented by professor of architecture Erno Rubik in 1974. It has approximately 43 quintillion different arrangments (status). To put this into perspective, if one had one standard-sized Rubik's Cube for each status, one could cover the Earth's surface 275 times, or stack them in a tower 261 light-years high.
The standard rubik cube has 6 faces, on each face has 9 fixed colors. The standard moves are R(turning clokewise) and R'(turn-counterclockwise) These two moves can be apploied to each face which generate different moves which are called R(right), R'(reverse-right), L(left),L'(reverse-left), U, U', B, B' and so on... Two Rubik cube computer models developed for 3 * 3 and N * N cubes.
The Rubik cube open schematic along with a matrix is shown below. The matrix is called the superposition matrix (M') which saves the information from previous moves in it (M).
The relations between M and M' for second face of Rubik cube are seen at the following handout.
The cube faces are numbered (between 1 to 6) along with the digits (1 to 9 = 3 * 3) . The movement of digits on each face and the faces which are engaged are seen in the below handout.
The program steps are provided at the following:
- Generate a random integer between 1 and 6 (NumOfFaces).
- Chose the face randomly that the move gonna be committed on.
- Generate a random integer between 1 and 2. The 1 clockwise turns and 2 is counterclockwise.
- Committee moves on the Rubik cube Face and updates the superposition matrix. Or you may apply your favorite move and do sensitivity analysis The Moves are committed based on the Face and move direction.
The first building block of the program is provided at the following. A similar procedure is taken for the rest 5 Faces of Rubik cube.
Commit moves on Face_1.
if (Face ==1){
if (Move ==1){
#update face corners
M_SuperPosition[1,1] <- M[1,7]
M_SuperPosition[1,3] <- M[1,1]
M_SuperPosition[1,9] <- M[1,3]
M_SuperPosition[1,7] <- M[1,9]
#update face edge
M_SuperPosition[1,2] <- M[1,4]
M_SuperPosition[1,6] <- M[1,2]
M_SuperPosition[1,8] <- M[1,6]
M_SuperPosition[1,4] <- M[1,8]
#update Edge#1
M_SuperPosition[5,7] <- M[2,9]
M_SuperPosition[4,1] <- M[5,7]
M_SuperPosition[3,3] <- M[4,1]
M_SuperPosition[2,9] <- M[3,3]
#update Edge#2
M_SuperPosition[5,8] <- M[2,6]
M_SuperPosition[4,4] <- M[5,8]
M_SuperPosition[3,2] <- M[4,4]
M_SuperPosition[2,6] <- M[3,2]
#update Edge#3
M_SuperPosition[5,9] <- M[2,3]
M_SuperPosition[4,7] <- M[5,9]
M_SuperPosition[3,1] <- M[4,7]
M_SuperPosition[2,3] <- M[3,1]
}
else {
#update face corners
M_SuperPosition[1,1] <- M[1,3]
M_SuperPosition[1,3] <- M[1,9]
M_SuperPosition[1,9] <- M[1,7]
M_SuperPosition[1,7] <- M[1,1]
#update face edge
M_SuperPosition[1,2] <- M[1,6]
M_SuperPosition[1,6] <- M[1,8]
M_SuperPosition[1,8] <- M[1,4]
M_SuperPosition[1,4] <- M[1,2]
#update Edge#1
M_SuperPosition[5,7] <- M[4,1]
M_SuperPosition[4,1] <- M[3,3]
M_SuperPosition[3,3] <- M[2,9]
M_SuperPosition[2,9] <- M[5,7]
#update Edge#2
M_SuperPosition[5,8] <- M[4,4]
M_SuperPosition[4,4] <- M[3,2]
M_SuperPosition[3,2] <- M[2,6]
M_SuperPosition[2,6] <- M[5,8]
#update Edge#3
M_SuperPosition[5,9] <- M[4,7]
M_SuperPosition[4,7] <- M[3,1]
M_SuperPosition[3,1] <- M[2,3]
M_SuperPosition[2,3] <- M[5,9]
}
#update M
M <- M_SuperPosition
}
Once the moves are commited the resuls are ploted in the following figure.
The program steps are provided at the following:
-
Create two types of the matrix for each Face.
-
One is called the base matrix (F[i]) and another one is the superposition matrix (FS[i]) which takes the moves on each Rubik's cube faces and memorizes and passes it to the base matrix (F[i]).
-
Create two types of the matrix for each Face. One is called the base matrix (F[i]) and another one is the superposition matrix (FS[i]) which takes the moves on each Rubik's cube faces and memorizes it and pass it to the base matrix (F[i]).
-
number of cube faces which is 6 for 3 * 3 rubik cube. For shapes with more faces (more than 6) this gigit should be opened and investigated! This digit acts like a cofficent of a model in determinstic models. However, in fact it has lot of independant parameters in it.
The first building block of the program is provided at the following. A similar procedure is taken for the rest 5 Faces of Rubik cube.
#Face#1, Rotate clockwise(Move=1) or counterclockwise(Move=2)
if (Face_Direction ==1){
if (layer_number==1){
if (Move ==1){
for (i in 1:floor(N/2)){
for (j in i:(N-i)){
FS1[i,j] <- F1[N+1-i,i]
FS1[j,N+1-i] <- F1[i,j]
FS1[N+1-i,N+1-i] <- F1[j,N+1-i]
FS1[N+1-i,i] <- F1[N+1-i,N+1-i]
}
}
}
else {
for (i in 1:floor(N/2)){
for (j in i:(N-i)){
FS1[i,j] <- F1[j,N+1-i]
FS1[j,N+1-i] <- F1[N+1-i,N+1-i]
FS1[N+1-i,N+1-i] <- F1[N+1-i,i]
FS1[N+1-i,i] <- F1[i,j]
}
}
}
}
#update other face#1 neighbors if move is 1
if (Move==1){
FS2[1:N,N+1-layer_number] <- F3[1,1:N]
FS5[N+1-layer_number,1:N] <- F2[1:N,N+1-layer_number]
FS4[1:N,layer_number] <- F5[N+1-layer_number,1:N]
FS3[layer_number,1:N] <- F4[1:N,layer_number]
}
else {
FS2[1:N,N+1-layer_number] <- F5[N+1-layer_number,1:N]
FS5[N+1-layer_number,1:N] <- F4[1:N,layer_number]
FS4[1:N,layer_number] <- F3[1,1:N]
FS3[layer_number,1:N] <- F2[1:N,N+1-layer_number]
}
#update base matrixs#1
F1 <- FS1
F2 <- FS2
F3 <- FS3
F4 <- FS4
F5 <- FS5
F6 <- FS6
}
A simulation for 5 * 5 Rubik cube after 10 randomly moves is provided at the following.
and another representation is as follow:
