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。

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

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 npimport 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 facedef 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 55 2 3 1 2 1 2 4 0 4 0 01 2 3 1 4 5 1 3 1 4 5 22 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 02 2 2 4 4 4 3 3 3 5 5 52 2 2 4 4 4 3 3 3 5 5 52 2 2 4 4 4 3 3 3 5 5 5 1 1 1 1 1 1 1 1 1Process finished with exit code 0

注：大寫為順時針，小寫為逆時針

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