# 群論與魔方：超級魔方還原的補充說明(上)

## 奇數階超級魔方的絕對／相對奇偶性

「絕對奇偶性」和「相對奇偶性」是「超級魔方」的重要性質，與「超級魔方」相關的四條重要定理(重列於下)都要用到這兩個概念，所以有必要對這兩個概念作更詳細的說明(在以下各式中，R-Parity和A-Parity分別代表「相對奇偶性」和「絕對奇偶性」；Ci、C、E0和C0分別代表「四角中心塊」、角塊、「正中邊塊」和「正中中心塊」軌道；P1和P2代表配成一對的「配對中心塊」軌道，M1和M2代表與P1、P2處於相同「非正中夾心層」的「十字中心塊」軌道；Ei和Mi分別代表處於同一「非正中夾心層」的「非正中邊塊」和「十字中心塊」軌道)：

00 (正中中心塊)
A-f, E-u, I-r, M-l, Q-d, U-b
02-06 (非正中邊塊)
AE-fu1, BL-fr5, CR-fd5, DO-fl1, EI-ru1, FD-fu5, GN-lu5, HW-bu1, IA-fr1, JH-ru5, KV-br5. LS-rd1, MQ-ld5, NX-bl1, OF-lu1, PC-fl5, QU-bd5, RP-ld1, SB-fd1, TK-rd5, UM-bl5, VT-bd1, WJ-br1, XG-bu5
03 (角塊)
AEI-fru, BLS-frd, CPR-fld, DFO-flu, GNX-blu, HJW-bru, KTV-brd, MQU-bld
05 (四角中心塊)
A-f11, B-f51, C-f55, D-f15, E-u11, F-u51, G-u55, H-u15, I-r11, J-r51, K-r55, L-r15, M-l55, N-l15, O-l11, P-l51, Q-d55, R-d15, S-d11, T-d51, U-b55, V-b15, W-b11, X-b51
08 (配對中心塊)
A-f21, B-f52, C-f45, D-f14, E-u21, F-u52, G-u45, H-u14, I-r21, J-r52, K-r45, L-r14, M-l45, N-l14, O-l21, P-l52, Q-d45, R-d14, S-d21, T-d52, U-b45, V-b14, W-b21, X-b52
11 (十字中心塊)
A-f31, B-f53, C-f35, D-f13, E-u31, F-u53, G-u35, H-u13, I-r31, J-r53, K-r35, L-r13, M-l35, N-l13, O-l31, P-l53, Q-d35, R-d13, S-d31, T-d53, U-b35, V-b13, W-b31, X-b53
12 (正中邊塊)
AL-fr3, BR-fd3, CO-fl3, DE-fu3, FN-lu3, GW-bu3, HI-ru3, JV-br3, KS-rd3, MX-bl3, PQ-ld3, TU-bd3

RmFm−1Rm−1Fm = (f, u, r)(b, d, l)     (1)

02-06 (非正中邊塊)
(fr1, ru1, fu1)(fr5, ru5, fu5)(fl1, rd1, bu1)(fl5, rd5, bu5)(fd1, br1, lu1)(fd5, br5, lu5)(bl1, ld1, bd1)(bl5, ld5, bd5)
0
03 (角塊)
(frd, bru, flu)(fld, brd, blu)
0
05 (四角中心塊)
(f11, r11, u11)(f15, r15, u15)(f51, r51, u51)(f55, r55, u55)(b11, l11, d11)(b15, l15, d15)(b51, l51, d51)(b55, l55, d55)
0
08 (配對中心塊)
(f14, r14, u14)(f21, r21, u21)(f45, r45, u45)(f52, r52, u52)(b14, l14, d14)(b21, l21, d21)(b45, l45, d45)(b52, l52, d52)
0
11 (十字中心塊)
(f13, r13, u13)(f31, r31, u31)(f35, r35, u35)(f53, r53, u53)(b13, l13, d13)(b31, l31, d31)(b35, l35, d35)(b53, l53, d53)
0
12 (正中邊塊)
(fr3, ru3, fu3)(fl3, rd3, bu3)(fu3, fr3, ru3)(fd3, br3, lu3)
0

RmFm−1Rm−1Fm
(f, u, r)(b, d, l)
(1)
RmFmRm−1Fm−1
(f, u, l)(b, d, r)
(2)
Rm−1FmRmFm−1
(f, d, r)(b, u, l)
(3)
Rm−1Fm−1RmFm
(f, d, l)(b, u, r)
(4)
Fm−1RmFmRm−1
(f, r, u)(b, l, d)
(5)
FmRmFm−1Rm−1
(f, l, u)(b, r, d)
(6)
FmRm−1Fm−1Rm
(f, r, d)(b, l, u)
(7)
Fm−1Rm−1FmRm
(f, l, d)(b, r, u)
(8)

(1) • (2) = (f, l, d)(b, r, u)

(1) • (5) = (f)(b)(r)(l)(u)(d)

(1) • (6) = (r, l)(u, d)

(1) • (6)
(r, l)(u, d)
(9)
(1) • (7)
(f, b)(u, d)
(10)
(1) • (8)
(f, b)(r, l)
(11)

ODD = (f, r, b, l)     (12)

02-06 (非正中邊塊)
(fr1, fl1, bl1, br1)(fr5, fl5, bl5, br5)(fu1, lu1, bu5, ru5)(fu5, lu5, bu1, ru1)(fd1, ld1, bd5, rd5)(fd5, ld5, bd1, rd1)
0
03 (角塊)
(fru, flu, blu, bru)(frd, fld, bld, brd)
0
05 (四角中心塊)
(f11, l11, b51, r51)(f15, l15, b11, r11)(f51, l51, b55, r55)(f55, l55, b15, r15)(u11, u51, u55, u15)(d11, d15, d55, d51)
0
08 (配對中心塊)
(f14,l14, b21, r21)(f21, l21, b52, r52)(f45, l45, b14, r14)(f52, l52, b45, r45)(u14, u21, u52, u45)(d14, d45, d52, d21)
0
11 (十字中心塊)
(f13, l13, b31, r31)(f31, l31, b53, r53)(f35, l35, b13, r13)(f53, l53, b35, r35)(u13, u31, u53, u35)(d13, d35, d53, d31)
0
12 (正中邊塊)
(fr3, fl3, bl3, br3)(fu3, lu3, bu3, ru3)(fd3, ld3, bd3, rd3)
1

(f, u)(b, d)(r, l) = (2) • ODD

 L−1L1−1L2−1 RR1R2 F−1F1−1F2−1 BB1B2 U−1U1−1U2−1 DD1D2 LnL1nL2n D−1D1−1D2−1 UU1U2 B−1B1−1B2−1 FF1F2 R−1R1−1R2−1 LL1L2 U−nU1−nU2−n = (u3n)(ln) (12) (UU1U2 RR1R2 LL1L2 U2U12U22 R−1R1−1R2−1 L−1L1−1L2−1)2 = (u2) (13)

05 (四角中心塊)
(u11, u51, u55, u15)(l11, l15, l55, l51)
0
08 (配對中心塊)
(u14, u21, u52, u45)(l14, l45, l52, l21)
0
11 (十字中心塊)
(u13, u31, u53, u35)(l13, l35, l53, l31)
0

## 偶數階超級魔方的絕對／相對奇偶性

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