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思路

復原魔方困難問題的分解：

1、用合適的數據結構表示出三階魔方的六個面以及每一面的顏色

2、每一次不同旋轉操作的實現

3、復原時如何判斷當前魔方應該使用哪一種公式

本次實驗實現了前兩個操作，具體思路是：

用numpy庫中的矩陣將六個面分別表示出來，按上下左右前后的順序放入列表中。再依據流行公式里的方法編寫對每一個面進行旋轉操作的函數，調用函數實現魔方的旋轉。最后輸入指令可得到旋轉之后的魔方，以及執行逆序指令后驗證魔方還原。

預備知識

矩陣：使用numpy庫中的矩陣結構

函數說明：

U: 上面順時針旋轉 90°

D: 底面順時針旋轉 90°

L: 左面順時針旋轉 90°

R: 右面順時針旋轉 90°

F: 正面順時針旋轉 90°

B: 背面順時針旋轉 90°

**注：**字母前加上下劃線 ‘_' 表示逆時針

代碼詳解

本次實驗將【上、下、左、右、前、后】六個面用數字【0、1、2、3、4、5】表示原本每個面的顏色，并依次存入列表faces【】里（即：faces[0]中存放的是最上面的數字全為0的三階矩陣）

注：魔方視角始終固定，即在整個過程中正（左…）面始終是正（左…）面

# 創建六個面，放在faces列表里，順序為上（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])

每一個面的順時針和逆時針旋轉由函數 clockwise() 和 antiClockwise() 實現

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

# 該面順時針旋轉 90 度
def clockwise(face):
    face = face.transpose().dot(t)
    return face

# 該面逆時針旋轉 90 度
def antiClockwise(face):
    face = face.dot(t).transpose()
    return face

A.transpose() 方法是實現 A 矩陣的轉置

A.dot(B) 方法是實現 A乘以矩陣B

通過計算，上述方法可以實現矩陣順時針或者逆時針旋轉的效果

在這里以左面的順時針旋轉 90°為例，其它旋轉方式可以類比

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、直接調用函數將左面（第2面）順時針旋轉 90°

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

2、這里采用深度復制，使用 cp.deepcopy() 的方法，避免直接使用等號 ‘=' 導致不同的變量指向同一個值。這時，【e、f、g、h】和【a、b、c、d】代表魔方的

【正面、底面順時針旋轉90°、背面逆時針旋轉90°、上面順時針旋轉90°】

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

旋轉的目的是：

在左面旋轉的過程中，左面會影響到其它四個面，但對其它四個面的影響是不同的。例如正面、底面和上面被影響的是第一列，而背面被影響的是第三列。我們為了使各面統一起來，方便數值的改變，我們選擇將正、底、上面順時針旋轉90°，將背面逆時針旋轉90°。這時，我們只需按順序交換每一面的第一行，最后再逆時針或順時針轉回來即可。

3、按順序交換：正面第一行傳遞到底面第一行

上面第一行傳遞到正面第一行

背面第一行傳遞到上面第一行

底面第一行傳遞到背面第一行

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

最后再依次根據上述操作逆旋轉回去：

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

代碼

import numpy as np
import copy as cp

# 創建六個面，放在faces列表里，順序為上（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]])

# 該面順時針旋轉 90 度
def clockwise(face):
    face = face.transpose().dot(t)
    return face

# 該面逆時針旋轉 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("請輸入步驟:")
moves(faces, lst)
print("執行后的魔方為")
toString(faces)
reverse = ''.join(map(chr, map(lambda x: ord(x) ^ 32, lst)))[::-1]
moves(faces, reverse)
print("魔方恢復步驟：", reverse)
toString(faces)

示例

請輸入步驟:UBLDFRULFDRULBGBVFDRLLBFLLDSSDBVDJFRUDLRFBDLFBbdj
執行后的魔方為

      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

魔方恢復步驟： 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：大寫為順時針，小寫為逆時針

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