python实现三阶魔方还原

思路

复原魔方困难问题的分解：

​ 1、用合适的数据结构表示出三阶魔方的六个面以及每一面的颜色

​ 2、每一次不同旋转操作的实现

​ 3、复原时如何判断当前魔方应该使用哪一种公式

本次实验实现了前两个操作，具体思路是：

用numpy库中的矩阵将六个面分别表示出来，按上下左右前后的顺序放入列表中。再依据流行公式里的方法编写对每一个面进行旋转操作的函数，调用函数实现魔方的旋转。最后输入指令可得到旋转之后的魔方，以及执行逆序指令后验证魔方还原。

预备知识

• 矩阵：使用numpy库中的矩阵结构

• 函数说明：

  U: 上面顺时针旋转 90°

  D: 底面顺时针旋转 90°

  L: 左面顺时针旋转 90°

  R: 右面顺时针旋转 90°

  F: 正面顺时针旋转 90°

  B: 背面顺时针旋转 90°

  **注：**字母前加上下划线 ‘_’ 表示逆时针

代码详解

• 本次实验将【上、下、左、右、前、后】六个面用数字【0、1、2、3、4、5】表示原本每个面的颜色，并依次存入列表faces【】里（即：faces[0]中存放的是最上面的数字全为0的三阶矩阵）

  注：魔方视角始终固定，即在整个过程中正（左…）面始终是正（左…）面

  # 创建六个面，放在faces列表里，顺序为上（0），下（1），左（2），右（3），前（4），后（5）
  faces = [np.zeros((3, 3))]
  
  for i in range(1, 6):
      faces.append(np.ones((3, 3)) + faces[i - 1])
  

  

• 每一个面的 顺时针 和 逆时针 旋转由函数 clockwise() 和 antiClockwise() 实现

  t = np.array([[0, 0, 1],
                [0, 1, 0],
                [1, 0, 0]])
  
  # 该面顺时针旋转 90 度
  def clockwise(face):
      face = face.transpose().dot(t)
      return face
  
  # 该面逆时针旋转 90 度
  def antiClockwise(face):
      face = face.dot(t).transpose()
      return face
  

  A.transpose() 方法是实现 A 矩阵的转置

  A.dot(B) 方法是实现 A乘以矩阵B

  通过计算，上述方法可以实现矩阵顺时针或者逆时针旋转的效果

• 在这里以左面的顺时针旋转 90°为例，其它旋转方式可以类比

  def L(FACES):
      FACES[2] = clockwise(FACES[2])
      FACES_new = cp.deepcopy(FACES)
      a, b, c, d = clockwise(FACES_new[4]), clockwise(FACES_new[1]), antiClockwise(FACES_new[5]), clockwise(FACES_new[0])
      e, f, g, h = cp.deepcopy(a), cp.deepcopy(b), cp.deepcopy(c), cp.deepcopy(d)
      e[0], f[0], g[0], h[0] = d[0], a[0], b[0], c[0]
      FACES[4], FACES[1], FACES[5], FACES[0] = antiClockwise(e), antiClockwise(f), clockwise(g), antiClockwise(h)
  

  1、直接调用函数将左面（第2面）顺时针旋转 90°

  FACES[2] = clockwise(FACES[2])
  

  2、这里采用深度复制，使用 cp.deepcopy() 的方法，避免直接使用等号 ‘=’ 导致不同的变量指向同一个值。这时，【e、f、g、h】和【a、b、c、d】代表魔方的

  【正面、底面顺时针旋转90°、背面逆时针旋转90°、上面顺时针旋转90°】

  a, b, c, d = clockwise(FACES_new[4]), clockwise(FACES_new[1]), antiClockwise(FACES_new[5]), clockwise(FACES_new[0])
  

  旋转的目的是：

  在左面旋转的过程中，左面会影响到其它四个面，但对其它四个面的影响是不同的。例如正面、底面和上面被影响的是第一列，而背面被影响的是第三列。我们为了使各面统一起来，方便数值的改变，我们选择将正、底、上面顺时针旋转90°，将背面逆时针旋转90°。这时，我们只需按顺序交换每一面的第一行，最后再逆时针或顺时针转回来即可。

  

  3、按顺序交换：正面第一行传递到底面第一行

  ​ 上面第一行传递到正面第一行

  ​ 背面第一行传递到上面第一行

  ​ 底面第一行传递到背面第一行

  e[0], f[0], g[0], h[0] = d[0], a[0], b[0], c[0]
  

  最后再依次根据上述操作逆旋转回去：

  FACES[4], FACES[1], FACES[5], FACES[0] = antiClockwise(e), antiClockwise(f), clockwise(g), antiClockwise(h)
  

代码

import numpy as np
import copy as cp

# 创建六个面，放在faces列表里，顺序为上（0），下（1），左（2），右（3），前（4），后（5）
faces = [np.zeros((3, 3))]

for i in range(1, 6):
    faces.append(np.ones((3, 3)) + faces[i - 1])

t = np.array([[0, 0, 1],
              [0, 1, 0],
              [1, 0, 0]])

# 该面顺时针旋转 90 度
def clockwise(face):
    face = face.transpose().dot(t)
    return face

# 该面逆时针旋转 90 度
def antiClockwise(face):
    face = face.dot(t).transpose()
    return face


def U(FACES):
    FACES[0] = clockwise(FACES[0])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = FACES_new[4], FACES_new[2], FACES_new[5], FACES_new[3]
    FACES[4][0], FACES[2][0], FACES[5][0], FACES[3][0] = d[0], a[0], b[0], c[0]


def _U(FACES):
    FACES[0] = antiClockwise(FACES[0])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = FACES_new[4], FACES_new[2], FACES_new[5], FACES_new[3]
    FACES[4][0], FACES[2][0], FACES[5][0], FACES[3][0] = b[0], c[0], d[0], a[0]


def U2(FACES):
    for i in range(2):
        U(FACES)
    '''
    FACES[0] = clockwise(clockwise(FACES[0]))
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = FACES_new[4], FACES_new[2], FACES_new[5], FACES_new[3]
    FACES[4][0], FACES[2][0], FACES[5][0], FACES[3][0] = c[0], d[0], a[0], b[0]
    '''


def D(FACES):
    FACES[1] = clockwise(FACES[1])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = FACES_new[4], FACES_new[2], FACES_new[5], FACES_new[3]
    FACES[4][2], FACES[2][2], FACES[5][2], FACES[3][2] = b[2], c[2], d[2], a[2]


def _D(FACES):
    FACES[1] = antiClockwise(FACES[1])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = FACES_new[4], FACES_new[2], FACES_new[5], FACES_new[3]
    FACES[4][2], FACES[2][2], FACES[5][2], FACES[3][2] = d[2], a[2], b[2], c[2]


def D2(FACES):
    for i in range(2):
        D(FACES)
    '''
    FACES[1] = clockwise(clockwise(FACES[1]))
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = FACES_new[4], FACES_new[2], FACES_new[5], FACES_new[3]
    FACES[4][2], FACES[2][2], FACES[5][2], FACES[3][2] = c[2], d[2], a[2], b[2]
    '''


def L(FACES):
    FACES[2] = clockwise(FACES[2])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = clockwise(FACES_new[4]), clockwise(FACES_new[1]), antiClockwise(FACES_new[5]), clockwise(FACES_new[0])
    e, f, g, h = cp.deepcopy(a), cp.deepcopy(b), cp.deepcopy(c), cp.deepcopy(d)
    e[0], f[0], g[0], h[0] = d[0], a[0], b[0], c[0]
    FACES[4], FACES[1], FACES[5], FACES[0] = antiClockwise(e), antiClockwise(f), clockwise(g), antiClockwise(h)


def _L(FACES):
    FACES[2] = antiClockwise(FACES[2])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = clockwise(FACES_new[4]), clockwise(FACES_new[1]), antiClockwise(FACES_new[5]), clockwise(FACES_new[0])
    e, f, g, h = cp.deepcopy(a), cp.deepcopy(b), cp.deepcopy(c), cp.deepcopy(d)
    e[0], f[0], g[0], h[0] = b[0], c[0], d[0], a[0]
    FACES[4], FACES[1], FACES[5], FACES[0] = antiClockwise(e), antiClockwise(f), clockwise(g), antiClockwise(h)


def L2(FACES):
    for i in range(2):
        L(FACES)


# 上（0），下（1），左（2），右（3），前（4），后（5）
def R(FACES):
    FACES[3] = clockwise(FACES[3])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = antiClockwise(FACES_new[4]), antiClockwise(FACES_new[1]), clockwise(FACES_new[5]), antiClockwise(
        FACES_new[0])
    e, f, g, h = cp.deepcopy(a), cp.deepcopy(b), cp.deepcopy(c), cp.deepcopy(d)
    g[0], f[0], e[0], h[0] = d[0], c[0], b[0], a[0]
    FACES[4], FACES[1], FACES[5], FACES[0] = clockwise(e), clockwise(f), antiClockwise(g), clockwise(h)


def _R(FACES):
    FACES[3] = antiClockwise(FACES[3])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = antiClockwise(FACES_new[4]), antiClockwise(FACES_new[1]), clockwise(FACES_new[5]), antiClockwise(
        FACES_new[0])
    e, f, g, h = cp.deepcopy(a), cp.deepcopy(b), cp.deepcopy(c), cp.deepcopy(d)
    f[0], g[0], h[0], e[0] = a[0], b[0], c[0], d[0]
    FACES[4], FACES[1], FACES[5], FACES[0] = clockwise(e), clockwise(f), antiClockwise(g), clockwise(h)


def R2(FACES):
    for i in range(2):
        R(FACES)


def F(FACES):
    FACES[4] = clockwise(FACES[4])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = clockwise(clockwise(FACES_new[0])), FACES_new[1], antiClockwise(FACES_new[2]), clockwise(FACES_new[3])
    e, f, g, h = cp.deepcopy(a), cp.deepcopy(b), cp.deepcopy(c), cp.deepcopy(d)
    e[0], g[0], f[0], h[0] = c[0], b[0], d[0], a[0]
    FACES[0], FACES[1], FACES[2], FACES[3] = clockwise(clockwise(e)), f, clockwise(g), antiClockwise(h)


def _F(FACES):
    FACES[4] = antiClockwise(FACES[4])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = clockwise(clockwise(FACES_new[0])), FACES_new[1], antiClockwise(FACES_new[2]), clockwise(FACES_new[3])
    e, f, g, h = cp.deepcopy(a), cp.deepcopy(b), cp.deepcopy(c), cp.deepcopy(d)
    g[0], f[0], h[0], e[0] = a[0], c[0], b[0], d[0]
    FACES[0], FACES[1], FACES[2], FACES[3] = clockwise(clockwise(e)), f, clockwise(g), antiClockwise(h)


def F2(FACES):
    for _ in range(2):
        F(FACES)


# 上（0），下（1），左（2），右（3），前（4），后（5）
def B(FACES):
    FACES[5] = clockwise(FACES[5])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = FACES_new[0], clockwise(clockwise(FACES_new[1])), clockwise(FACES_new[2]), antiClockwise(FACES_new[3])
    e, f, g, h = cp.deepcopy(a), cp.deepcopy(b), cp.deepcopy(c), cp.deepcopy(d)
    g[0], f[0], h[0], e[0] = a[0], c[0], b[0], d[0]
    FACES[0], FACES[1], FACES[2], FACES[3] = e, clockwise(clockwise(f)), antiClockwise(g), clockwise(h)


def _B(FACES):
    FACES[5] = antiClockwise(FACES[5])
    FACES_new = cp.deepcopy(FACES)
    a, b, c, d = FACES_new[0], clockwise(clockwise(FACES_new[1])), clockwise(FACES_new[2]), antiClockwise(FACES_new[3])
    e, f, g, h = cp.deepcopy(a), cp.deepcopy(b), cp.deepcopy(c), cp.deepcopy(d)
    e[0], g[0], f[0], h[0] = c[0], b[0], d[0], a[0]
    FACES[0], FACES[1], FACES[2], FACES[3] = e, clockwise(clockwise(f)), antiClockwise(g), clockwise(h)


def B2(FACES):
    for i in range(2):
        B(FACES)


'''
                          |************|
                          |*U1**U2**U3*|
                          |************|
                          |*U4**U5**U6*|
                          |************|
                          |*U7**U8**U9*|
                          |************|
              ************|************|************|************|
              *L1**L2**L3*|*F1**F2**F3*|*R1**R2**R3*|*B1**B2**B3*|
              ************|************|************|************|
              *L4**L5**L6*|*F4**F5**F6*|*R4**R5**R6*|*B4**B5**B6*|
              ************|************|************|************|
              *L7**L8**L9*|*F7**F8**F9*|*R7**R8**R9*|*B7**B8**B9*|
              ************|************|************|************|
                          |************|
                          |*D1**D2**D3*|
                          |************|
                          |*D4**D5**D6*|
                          |************|
                          |*D7**D8**D9*|
                          |************|
'''


def toString(FACES):
    print()
    for i in range(3):
        print("     ", int(FACES[0][i][0]), int(FACES[0][i][1]), int(FACES[0][i][2]))
    for i in range(3):
        print(int(FACES[2][i][0]), int(FACES[2][i][1]), int(FACES[2][i][2]), end=" ")
        print(int(FACES[4][i][0]), int(FACES[4][i][1]), int(FACES[4][i][2]), end=" ")
        print(int(FACES[3][i][0]), int(FACES[3][i][1]), int(FACES[3][i][2]), end=" ")
        print(int(FACES[5][i][0]), int(FACES[5][i][1]), int(FACES[5][i][2]))
    for i in range(3):
        print("     ", int(FACES[1][i][0]), int(FACES[1][i][1]), int(FACES[1][i][2]))
    print()


def moves(FACES, lst):
    for x in lst:
        if x == 'U':
            U(faces)
        elif x == 'u':
            _U(faces)
        elif x == 'D':
            D(faces)
        elif x == 'd':
            _D(faces)
        elif x == 'L':
            L(faces)
        elif x == 'l':
            _L(faces)
        elif x == 'R':
            R(faces)
        elif x == 'r':
            _R(faces)
        elif x == 'F':
            F(faces)
        elif x == 'f':
            _F(faces)
        elif x == 'B':
            B(faces)
        elif x == 'b':
            _B(faces)


lst = input("请输入步骤:")
moves(faces, lst)
print("执行后的魔方为")
toString(faces)
reverse = ''.join(map(chr, map(lambda x: ord(x) ^ 32, lst)))[::-1]
moves(faces, reverse)
print("魔方恢复步骤：", reverse)
toString(faces)

示例

请输入步骤:UBLDFRULFDRULBGBVFDRLLBFLLDSSDBVDJFRUDLRFBDLFBbdj
执行后的魔方为

      2 5 3
      5 0 2
      5 0 5
5 2 3 1 2 1 2 4 0 4 0 0
1 2 3 1 4 5 1 3 1 4 5 2
2 5 2 4 4 3 1 0 5 3 4 4
      1 0 4
      3 1 3
      0 3 0

魔方恢复步骤： JDBbfldbfrldurfjdvbdssdllfbllrdfvbgblurdflurfdlbu

      0 0 0
      0 0 0
      0 0 0
2 2 2 4 4 4 3 3 3 5 5 5
2 2 2 4 4 4 3 3 3 5 5 5
2 2 2 4 4 4 3 3 3 5 5 5
      1 1 1
      1 1 1
      1 1 1


Process finished with exit code 0

注：大写为顺时针，小写为逆时针

最后要感谢本次实验的主要贡献者：🐷 ~