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标题: 魔方难题（定期发，纯原创） [打印本页]

作者: brainyuan 时间: 2009-4-9 14:45:58 标题: 魔方难题（定期发，纯原创）

一个还原的魔方，对其进行如下6种操作：
U2
D2
F2
B2
L2
R2
要求每种操作必须要用，且只能用1次。使用顺序任意。
问，共能得到多少种不同的状态？

作者: yq_118 时间: 2009-4-9 14:59:27

期待高手给出答案。不过可以试下，不会超过6！=720.

[ 本帖最后由 yq_118 于 2009-4-9 15:00 编辑 ]

作者: conwood 时间: 2009-4-9 15:02:44

进一步的，{UD}, {FB}, {LR}会导致重复状态，可以从720里减去对应的数，楼下高手继续。

作者: gejunji 时间: 2009-4-9 15:45:23

这个不太理解。需要深入探究探究

作者: wrcgqhdf521 时间: 2009-4-9 16:17:50

很复杂啊！！

作者: magi 时间: 2009-4-9 16:39:38

只有一种，就是棋盘图案！棋盘的状态只有一种！楼主说对不对

作者: yq_118 时间: 2009-4-9 16:57:12

原帖由 magi 于 2009-4-9 16:39 发表
只有一种，就是棋盘图案！棋盘的状态只有一种！楼主说对不对

明显不会只有一种,这个真的很明显,图就不发了.

作者: 乌木 时间: 2009-4-9 17:11:09

原帖由 conwood 于 2009-4-9 15:02 发表
进一步的，{UD}, {FB}, {LR}会导致重复状态，可以从720里减去对应的数，楼下高手继续。

你的意思是不是指：两条步骤，其余4步一样，一条是U2 D2，另一条是D2 U2，这两条要精简掉一条。比如，U2 D2 L2 R2 F2 B2 和 D2 U2 L2 R2 F2 B2 ，结果一样。

作者: Cielo 时间: 2009-4-9 17:20:24

个人感觉，这题发到数学趣题区更合适

作者: kexin_xiao 时间: 2009-4-9 20:25:50

不太明白,排列组合问题?

作者: Atato 时间: 2009-5-2 16:57:17

我觉得这个不是数学问题...

作者: bardy 时间: 2009-5-9 17:38:21

Cube subgroups

Subgroups generated by single moves

Suppose not all possible cube moves are allowed, for example suppose that you only use moves of the Up and Right faces. You can mix a cube that way, and then try to solve it again using only those moves. Clearly not all cube positions can be reached, since the 2×3×3 block at the bottom left is never disturbed by the U and R moves. In the same way we might restrict ourselves to using only half turns. There are many choices as to which faces we allow, and whether each face uses only half turns or not. Below is a list containing all possibilities.
Generators Size Corners Edges Restrictions
0. - 1 = 1 * 1
1a. U2 2 = 2 * 2 / 2
1b. U 4 = 4 * 4 / 4
2a. U2,R2 12 = 3! * 3! 22/2 / 3!
2b. U,R2 14400 = 6!/6 * 5! 2! / 2
2c. U,R 73483200 = 6!/6 36/3 * 7! / 2
3a. U2, D2, F2 96 = 4! * 23 / 2
3b. U2, R2, F2 2592 = 4! * 3! 3! 3! / 2
3c. U, R2, F2 10886400 = 7! * 6! 3! / 2
3d. U, R2, L2 58060800 = 8! * 6! 2! 2! / 2
3e. U2, R, L2 58060800 = 8! * 6! 2! 2! / 2
3f. U, R2, D 1625702400 = 8! * 8! 2! / 2
3g. U2, R, F 666639590400 = 7! 37/3 * 9! / 2
3h. U, R, D2 3555411148800 = 8! 38/3 * 8! 2! / 2
3i. U, R, D 159993501696000 = 8! 38/3 * 10! / 2
3j. U, R, F 170659735142400 = 7! 37/3 * 9! 29/2 / 2
4a. U2, D2, F2, B2 192 = 4! * 4! 24/2 / 4!
4b. U2, D2, F2, R2 165888 = 4! 4 * 4! 4! 3! / 2
4c. U, R2, L2, D2 116121600 = 8! * 6! 2! 2! 2! / 2
4d. U2, D2, F2, R 2438553600 = 8! * 7! 4! / 2
4e. U, R2, L2, D 3251404800 = 8! * 8! 2! 2! / 2
4f. U, D2, F2, R2 4877107200 = 8! * 8! 3! / 2
4g. U, D, F2, R2 4877107200 = 8! * 8! 3! / 2
4h. U, D2, F2, R 1759928518656000 = 8! 38/3 * 11! / 2
4i. U, D, F2, R 1759928518656000 = 8! 38/3 * 11! / 2
4j. U2, D2, F, R 1759928518656000 = 8! 38/3 * 11! / 2
4k. U2, D, F, R 1802166803103744000 = 8! 38/3 * 11! 211/2 / 2
4l. U, D, F, R 1802166803103744000 = 8! 38/3 * 11! 211/2 / 2
5a. U2, F2, B2, R2, L2 663552 = 4! 4 * 4! 4! 4! / 2
5b. U, D2, F2, B2, R2 19508428800 = 8! * 8! 4! / 2
5c. U, D, F2, B2, R2 19508428800 = 8! * 8! 4! / 2
5d. U, F2, B2, R2, L2 19508428800 = 8! * 8! 4! / 2
5e. U2, F, B2, R, L2 21119142223872000 = 8! 38/3 * 12! / 2
5f. U2, F, B2, R, L 21119142223872000 = 8! 38/3 * 12! / 2
5g. U2, F, B, R, L 21119142223872000 = 8! 38/3 * 12! / 2
5h. U2, D2, F, B, R 21119142223872000 = 8! 38/3 * 12! / 2
5i. U2, D2, F, B2, R 21119142223872000 = 8! 38/3 * 12! / 2
5j. U, F, B2, R, L2 43252003274489856000 = 8! 38/3 * 12! 212/2 / 2
5k. U, F, B, R, L2 43252003274489856000 = 8! 38/3 * 12! 212/2 / 2
5l. U, F, B, R, L 43252003274489856000 = 8! 38/3 * 12! 212/2 / 2
6a. U2, D2, F2, B2, R2, L2 663552 = 4! 4 * 4! 4! 4! / 2 / 1
6b. U, D, F2, B2, R2, L2 19508428800 = 8! * 8! 4! / 2
6c. U, D2, F2, B2, R2, L2 19508428800 = 8! * 8! 4! / 2
6d. U2, D2, F, B, R, L 21119142223872000 = 8! 38/3 * 12! / 2
6e. U2, D2, F, B2, R, L 21119142223872000 = 8! 38/3 * 12! / 2
6f. U2, D2, F, B2, R, L2 21119142223872000 = 8! 38/3 * 12! / 2
6g. U, D2, F, B2, R, L2 43252003274489856000 = 8! 38/3 * 12! 212/2 / 2
6h. U, D2, F, B, R, L2 43252003274489856000 = 8! 38/3 * 12! 212/2 / 2
6i. U, D2, F, B, R, L 43252003274489856000 = 8! 38/3 * 12! 212/2 / 2
6j. U, D, F, B, R, L 43252003274489856000 = 8! 38/3 * 12! 212/2 / 2

The list above is complete up to turns of the whole cube, e.g. <F,R> is not included because it is isomorphic to the group generated by U and R, which is on the list. I have also omitted the groups generated by R and L, R2 and L, or R2 and L2, because those trivially decompose.

The number of reachable positions (the size of the group) is shown, as well as the number of ways the corners or the edges can be arranged separately. The last column is a factor showing how much the edges and corners restrict each other. Usually this is a factor 2, due to the normal parity restriction that the parity of the total corner and edge permutation must be even. Thus once the corners have been solved the parity of the edges is forced, and they have half as many possible arrangements than they had before the corners were solved. In some rare cases the edges and corners influence each other by more than this.

Some of the groups listed are actually identical. This can easily be shown using the following two move sequences.
1. D = F2R2D2F2U2R2F2 U F2R2U2F2D2R2F2 (uses U, F2, R2, D2)
2. D2 = F2R2L2B2U2F2R2L2B2 (uses F2, B2, R2, L2, U2)
For example 4f=<U, D2, F2, R2> and 4g=<U, D, F2, R2> are the same because with the first move sequence you can use only U, D2, F2, and R2 to get the same effect as D (and conversely with D you can of course get D2). Similarly 4h, 4i are equal. In fact, using these two move sequences you can show that any two groups on the list of the same size are identical groups, except that 4j is slightly different from 4h/4i. In group 4j the edge orientations differ from those in 4h/4i, though in all other respects they are the same. Thus they are isomorphic groups.

If you wish to solve a position using the same set of moves you used to mix them, it is often easiest to solve the corners first, and then the edges. Below is a table of useful move sequences. With conjugates of these, all positions in the above groups can be solved. Note that the sequences for the corners may disturb edges.
Effect Generators Sequence
URF- URB+ U, R RU'RU'RU' R'UR'UR'U
(UFL, DFR, DBR) U, R2, F2 UF2U'R2UF2U'F2
(URF, DRB, DBL) U, R2, L2 UL2U'R2 UL2U'R2
(URF, DRB, DBL) U2, R, L2 R2U2R'L2U2R'L2U2RL2U2RL2U2
UF+ UR+ U, R, F RU'R2UFRUF2U'FR2F2R2F2
(UF, UR, UB) U, R RU2RURUR2U'R'U'R2
(UF, UR, UB) U, R2 U2R2U2R2 UR2UR2 U2R2U2R2 UR2UR2
(UF, UB)(RF, RB) U2, R2 R2U2 R2U2 R2U2
(FL, FR, BR) U2, R2, F2 F2U2R2U2 F2U2R2U2

Subgroups generated by slice/anti-slice moves

The groups generated by slice moves, or anti-slice moves are interesting because there are many nice patterns in these groups.
Generators Size Corners Edges Restrictions
a. U2D2, F2B2, R2L2 8 = 4 * 23 / 4
b. UD', FB', RL' 768 = 24 * 83/2 3!/2 / 24
c. UD, FB, RL 6144 = 4·24 * 43 3! 23/2 / 4!
d. UD, FB, RL, UD', FB', RL' 15925248 = 3! 4! 4 * 4! 4! 4! 3! 23/2 / 3!

The first of these, the slice-squared group, is trivial. The second, the slice group, is easy to understand if you consider the corners as fixed in space and the centres as moving pieces. Just line up each slice of edges with the corners, and you will be left with a spot pattern.

The third on the list is the anti-slice group. Positions in this group are more difficult to solve. First solve the corners relative to each other, which is in essence similar to solving the corners in the square group 6a. Next orient the edges using the move sequence RL UD FB RL UD FB, which flips the edges in the U/D layers. You should now have a cube with only opposing colours on each face. Line up the centres with the corners using squared (anti-)slices. The edges can be solved using the sequence RL UD F2B2 UD RL which is a 4H pattern on the sides.

The last group on the list is generated by all slice and anti-slice moves. Note that combining a slice and an anti-slice move you get a half turn of a single face, e.g. RL RL' = R2, so the square group (6a) is a subgroup of this one. It is relatively easy to solve by bringing it to a position in the square group. The edge flip in the previous paragraph is useful for this.

Various small subgroups

For many small finite groups, an isomorphic group can be found on the cube. For example a cyclic group of order 3 can be found by a 3-cycle of pieces. Below I will list some small groups and give generators on the cube that give such a group.
Group Generator
C2 Any half turn such as R2, or any edge flip.
C3 Any 3-cycle such as R2UD'F2U'D, or any corner twist.
C4 Any face turn such as R
C2×C2 R2L2 and F2B2
C5 Any 5-cycle, for example D2R2D'R2L2UR'L'B2RL'.
C6=C2×C3 R2F2
S3 R2F2R2F2 and R2
C7 Any 7-cycle
C8 Any 8-cycle (combined with a 2-cycle for parity reasons), e.g. R2F2 D2R2D2 B2R2 D R2L2 U F2B2, or flipped edge-4-cycle, e.g. L2D R2F2B2 L'R' F'L2F LR' F2B2U'.
C2×C4 R2, L
C2×C2×C2 R2L2, F2B2, U2D2.
D4 UD', and FB R2L2 F'B'
Q, Quaternion group i = (UR,UF)+(UL,UB)+ = R2 UR'U'R F R2 URU' F' UR2, j = (UB,UF)+(UR,UL)+ = L'U'B' U2 BLU FU2F', k = (UL,UF)+(UB,UR)+ = B'U'R U2 FRF'R' U' BRU'R'. Note that they all have order 4 and that ijk=jki=kij=ii=jj=kk.
A4 R2F2B2R2F2B2 and F2R2F2R2 gives all even permutations of the 4 corner columns.
C13 Impossible. 13 does not divide the order of the cube group.
C2×C2×C4 U2, D, Superflip.
C2×C2×C2×C2 U2D2, F2B2, R2L2, Superflip
D4×C2 UD', FB R2L2 F'B', and R2F2B2L2F2B2
C213 The 6H pattern F2R2L2B2L2R2 in its 6 possible orientations, the double corner swap U R2FR2 U2D2 L2BL2 UD2 in its 6 possible orientations, and 4-flip UF2D F2B2 DL2D F'B LB2L2 U'D F'R'L in its three possible orientations.

作者: bardy 时间: 2009-5-9 17:39:17

这个列表应该比较详细了，具体是否正确需要自己判断了

作者: lvyingru 时间: 2009-5-9 18:28:01

搞不懂。。

作者: 无为子 时间: 2009-7-9 06:16:37

高手大把，俺只来看看，正在学习中···ing

作者: noski 时间: 2009-7-23 23:36:18 标题: 回复 12# 的帖子

楼主的问题，U2、D2、F2、B2、R2、L2，这六个操作都使用一次，不同的组合，共有42个状态，程序求得，计算公式我给不出来。

另外，12楼的表格不错！
最少步区讨论的1x3x3魔方，正是三阶魔方的（U2,D2,F2,B2）子状态集，有192个状态。

---------------------------------
更正，这42的结果是不对的，只算了角块，没有算棱块。

[ 本帖最后由 noski 于 2009-7-24 18:23 编辑 ]

作者: migl 时间: 2009-7-24 16:26:45

U2 D2 F2 B2 L2 R2
这六个“数”进行排列组合，然后减去相同的状态数即可。

（我也感觉是不多于：6！=720，但又要少很多。）

比如：
U2 D2 F2 B2 L2 R2
D2 U2 F2 B2 L2 R2
U2 D2 F2 B2 L2 R2
U2 D2 B2 F2 L2 R2
U2 D2 F2 B2 L2 R2
U2 D2 F2 B2 R2 L2
这六种是同一个状态。（即：对称棋盘）
而：
U2 D2 F2 B2 L2 R2
U2 D2 L2 R2 F2 B2
F2 B2 U2 D2 L2 R2
F2 B2 L2 R2 U2 D2
L2 R2 U2 D2 F2 B2
L2 R2 F2 B2 U2 D2
都是“对称棋盘”。
所以至少有6*6=36条公式是“对称棋盘”，是同一种状态。

但是， L2 F2 R2 B2 U2 D2 却不是“对称棋盘”。

唉，本人脑细胞有限。等高手们继续发挥吧~~~

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