如何从数学角度证明魔方复原存在必可解策略？ 第1页

       ####2阶魔方 #矩阵中心对称 o <- function(A) {     t = A[1,1]     A[1,1] = A[2,2]     A[2,2] = t     t = A[1,2]     A[1,2] = A[2,1]     A[2,1] = t     A } #矩阵元素逆时针旋转 r <- function(A) {     t = A[1,1]     A[1,1] = A[1,2]     A[1,2] = A[2,2]     A[2,2] = A[2,1]     A[2,1] = t     A }  #矩阵元素顺时针旋转 v <- function(A) {     t = A[1,1]     A[1,1] = A[2,1]     A[2,1] = A[2,2]     A[2,2] = A[1,2]     A[1,2] = t     A }  #魔方的六个面 A = matrix(rep(1,4),2) B = matrix(rep(2,4),2) C = matrix(rep(3,4),2) a = matrix(rep(4,4),2) b = matrix(rep(5,4),2) c = matrix(rep(6,4),2)  #A为俯视图，B为正视图，C为右侧视图，X=A,B,C;结果为矩阵形式，方便绘图。 Xup <- function(A,a,B,b,C,c) {     Aup = matrix(rep(0,48),6)     Aup[3:4,5:6] = A     Aup[3:4,1:2] = a     Aup[1:2,5:6] = B     Aup[5:6,5:6] = b     Aup[3:4,3:4] = C     Aup[3:4,7:8] = c     Aup }  cube = list(A,a,B,b,C,c) cube[[7]] = Xup(A,a,B,b,C,c)   #B朝上的十字架展开图（A --> B），输入输出皆为列表   ABup <- function(Aup)     {     Bup = list()     Bup[[1]] = Aup[[1]]     Bup[[2]] = o(Aup[[2]])     Bup[[3]] = Aup[[3]]     Bup[[4]] = o(Aup[[4]])     Bup[[5]] = v(Aup[[5]])     Bup[[6]] = r(Aup[[6]])     Bup[[7]] = Xup(Bup[[3]],Bup[[4]],Bup[[2]],                    Bup[[1]],Bup[[5]],Bup[[6]])     Bup }  #（B --> A） BAup <- function(Bup)     {     Aup = list()     Aup[[1]] = Bup[[1]]     Aup[[2]] = o(Bup[[2]])     Aup[[3]] = Bup[[3]]     Aup[[4]] = o(Bup[[4]])     Aup[[5]] = r(Bup[[5]])     Aup[[6]] = v(Bup[[6]])     Aup[[7]] = Xup(Aup[[1]],Aup[[2]],Aup[[3]],                    Aup[[4]],Aup[[5]],Aup[[6]])     Aup }  #C朝上的十字架展开图（A --> C） ACup <- function(Aup)     {     Cup = list()     Cup[[1]] = Aup[[1]]     Cup[[2]] = Aup[[2]]     Cup[[3]] = r(Aup[[3]])     Cup[[4]] = v(Aup[[4]])     Cup[[5]] = Aup[[5]]     Cup[[6]] = Aup[[6]]     Cup[[7]] = Xup(Cup[[5]],Cup[[6]],Cup[[3]],                    Cup[[4]],Cup[[2]],Cup[[1]])     Cup }  #（C --> A） CAup <- function(Cup)     {     Aup = list()     Aup[[1]] = Cup[[1]]     Aup[[2]] = Cup[[2]]     Aup[[3]] = v(Cup[[3]])     Aup[[4]] = r(Cup[[4]])     Aup[[5]] = Cup[[5]]     Aup[[6]] = Cup[[6]]     Aup[[7]] = Xup(Aup[[1]],Aup[[2]],Aup[[3]],                    Aup[[4]],Aup[[5]],Aup[[6]])     Aup }    #画出魔方展开图 color = c("white","red","orange","yellow","green","blue","purple") par(mai=rep(0,4),oma=rep(0,4)) plot(0,0, type = "n", xlim = c(0,7), ylim = c(0,9)) draw <- function(Cube)    #输入矩阵 {     for(i in 1:6)for(j in 1:8)points(i, j, pch = 15, cex = 4, col = color[Cube[i,j]+1]) } draw(Xup(A,a,B,b,C,c))    #魔方初始状态  ##三大变换 #对A（红色面）逆时针旋转90度；此时List应该是A面为俯视面的形式 U <- function(List) {     unfold = Xup(List[[1]],List[[2]],List[[3]],                  List[[4]],List[[5]],List[[6]])     A = matrix(rep(0,48),6)     for(i in 2:5)for(j in 4:7) A[9-j,2+i] = unfold[i,j]    #坐标旋转由复数推导     unfold[2:5,4:7] = A[2:5,4:7]     A = unfold[3:4,5:6]     a = unfold[3:4,1:2]     B = unfold[1:2,5:6]     b = unfold[5:6,5:6]     C = unfold[3:4,3:4]     c = unfold[3:4,7:8]     List = list(A,a,B,b,C,c,unfold)     List    }  #U的逆变换，此时List应该是A面为俯视面的形式 V <- function(List) {     unfold = Xup(List[[1]],List[[2]],List[[3]],                  List[[4]],List[[5]],List[[6]])     A = matrix(rep(0,48),6)     for(i in 2:5)for(j in 4:7) A[j-2,9-i] = unfold[i,j]    #坐标旋转由复数推导     unfold[2:5,4:7] = A[2:5,4:7]     A = unfold[3:4,5:6]     a = unfold[3:4,1:2]     B = unfold[1:2,5:6]     b = unfold[5:6,5:6]     C = unfold[3:4,3:4]     c = unfold[3:4,7:8]     List = list(A,a,B,b,C,c,unfold)     List    }  #对B（橙色面）逆时针旋转90度；此时List应该是B面为俯视面的形式 F <- function(List) {     unfold = Xup(List[[3]],List[[4]],List[[2]],                  List[[1]],List[[5]],List[[6]])     A = matrix(rep(0,48),6)     for(i in 2:5)for(j in 4:7)A[9-j,2+i] = unfold[i,j]    #坐标旋转由复数推导     unfold[2:5,4:7] = A[2:5,4:7]     B = unfold[3:4,5:6]     b = unfold[3:4,1:2]     a = unfold[1:2,5:6]     A = unfold[5:6,5:6]     C = unfold[3:4,3:4]     c = unfold[3:4,7:8]     List = list(A,a,B,b,C,c,unfold)     List    }  #F的逆变换,此时List应该是B面为俯视面的形式 E <- function(List) {     unfold = Xup(List[[3]],List[[4]],List[[2]],                  List[[1]],List[[5]],List[[6]])     A = matrix(rep(0,48),6)     for(i in 2:5)for(j in 4:7)A[j-2,9-i] = unfold[i,j]    #坐标旋转由复数推导     unfold[2:5,4:7] = A[2:5,4:7]     B = unfold[3:4,5:6]     b = unfold[3:4,1:2]     a = unfold[1:2,5:6]     A = unfold[5:6,5:6]     C = unfold[3:4,3:4]     c = unfold[3:4,7:8]     List = list(A,a,B,b,C,c,unfold)     List    }  #对C（黄色面）逆时针旋转90度;此时List应该是C面为俯视面的形式 R <- function(List) {     unfold = Xup(List[[5]],List[[6]],List[[3]],                  List[[4]],List[[2]],List[[1]])     A = matrix(rep(0,48),6)     for(i in 2:5)for(j in 4:7)A[9-j,2+i] = unfold[i,j]    #坐标旋转由复数推导     unfold[2:5,4:7] = A[2:5,4:7]     C = unfold[3:4,5:6]     c = unfold[3:4,1:2]     B = unfold[1:2,5:6]     b = unfold[5:6,5:6]     A = unfold[3:4,7:8]     a = unfold[3:4,3:4]     List = list(A,a,B,b,C,c,unfold)     List    } #R的逆变换,此时List应该是C面为俯视面的形式 K <- function(List) {     unfold = Xup(List[[5]],List[[6]],List[[3]],                  List[[4]],List[[2]],List[[1]])     A = matrix(rep(0,48),6)     for(i in 2:5)for(j in 4:7)A[j-2,9-i] = unfold[i,j]    #坐标旋转由复数推导     unfold[2:5,4:7] = A[2:5,4:7]     C = unfold[3:4,5:6]     c = unfold[3:4,1:2]     B = unfold[1:2,5:6]     b = unfold[5:6,5:6]     a = unfold[3:4,3:4]     A = unfold[3:4,7:8]     List = list(A,a,B,b,C,c,unfold)     List    }   #魔方的变换 Transform <- function(Cha)  #输入由U/V/F/E/R/K构成的字符串 {     Cha = unlist(strsplit(Cha,split=""))    #将字符串的字母逐个拆开     for(i in Cha)     {         if(i=="U"){cube = U(cube)}         if(i=="V"){cube = V(cube)}         if(i=="F"){cube = F(ABup(cube)); cube = BAup(cube)}         if(i=="E"){cube = E(ABup(cube)); cube = BAup(cube)}         if(i=="R"){cube = R(ACup(cube)); cube = CAup(cube)}         if(i=="K"){cube = K(ACup(cube)); cube = CAup(cube)}         draw(cube[[7]])         Sys.sleep(0.2)     }     cube[[7]] }  Transform(rep("RVE",30))