How many stereoisomers does monofluoromonochloromonobromomonoiodocubane have? Optical isomers must be considered.
一氟一氯一溴一碘立方烷共有多少种同分异构体？需考虑旋光异构体。
6
7
10
20
28
32
38
48
57
64
70
75
82
88
93
100
6
7
10
20
28
32
38
48
57
64
70
75
82
88
93
100
First, consider the number of isomers when the four halogen atoms are identical:

(1) All 4X on the same face: 1,2,3,4- — 1 type  

(2) Three X on one face: cases include 1,2,3,5- / 1,2,3,6- / 1,2,4,5- — 3 types  

(3) Two X on the same face: cases include 1,2,5,6- / 1,3,5,7- — 2 types  

Thus, there are a total of 6 possible arrangements.  

<Checkpoint>[Four identical halogen atoms have 6 possible arrangements](1)</Checkpoint>  

Next, calculate the number of isomers for monofluoro-monochloro-monobromo-monoiodo cubane under these cases:  

(1) 1,2,3,4-: Fix the F atom, and there are 3 possibilities for the diagonal positions, each with optical isomers — total: 3×2=6 types  

<Checkpoint>[1,2,3,4- has 6 types](1)</Checkpoint>  

(2) 1,2,3,5-: Position 5 has 4 possibilities, position 1 has 3, all with optical isomers — total: 4×3×2=24 types  

<Checkpoint>[1,2,3,5- has 24 types](1)</Checkpoint>  

(3) 1,2,3,6-: Position 1 has 4 possibilities, all with optical isomers — total: 4×2=8 types  

<Checkpoint>[1,2,3,6- has 8 types](1)</Checkpoint>  

(4) 1,2,4,5-: The two adjacent atoms have $C_4^2$ possibilities, and the remaining two have 2 possibilities, all with optical isomers — total: 6×2×2=24 types  

<Checkpoint>[1,2,4,5- has 24 types](1)</Checkpoint>  

(5) 1,2,5,6-: Fix the F atom, adjacent positions have 3 possibilities, face-diagonal positions have 2, with no optical isomers — total: 3×2=6 types  

<Checkpoint>[1,2,5,6- has 6 types](1)</Checkpoint>  

(6) 1,3,5,7-: 2 optical isomers — 2 types  

<Checkpoint>[1,3,5,7- has 2 types](1)</Checkpoint>  

Thus, there are a total of 70 types.  

<Checkpoint>[There are 70 types of isomers in total](1)</Checkpoint>
先考虑4个卤原子相同时有几个异构体：

（1）4X在同一个面上：1, 2, 3, 4- 1种

（2）3X在一个面上：有1, 2, 3, 5-/1, 2, 3, 6-/1, 2, 4, 5- 3种情况

（3）2X在同一个面上：有1, 2, 5, 6-/1, 3, 5, 7- 2种

故共有6种排布情况

<Checkpoint>[四个相同的卤原子有6中排布情况](1)</Checkpoint>

接下来依次计算这些情况下一氟一氯一溴一碘立方烷的异构体数目：

（1）1,2,3,4-：固定F原子，对角有3种可能，且均有旋光异构体，共3×2=6种

<Checkpoint>[1,2,3,4-共有6种](1)</Checkpoint>

（2）1,2,3,5-：5号位有4种可能，1号位3种，均有旋光异构体，共4×3×2=24种

<Checkpoint>[1,2,3,5-共有24种](1)</Checkpoint>

（3）1,2,3,6-：1号位有4种可能，均有旋光异构体，共4×2=8种

<Checkpoint>[1,2,3,6-共有8种](1)</Checkpoint>

（4）1,2,4,5-：共边的两个原子有$C_4^2$种可能，剩下两个原子有2种可能，均有旋光异构体，共6×2×2=24种

<Checkpoint>[1,2,4,5-共有24种](1)</Checkpoint>

（5）1,2,5,6-：固定F原子，邻位有3种可能，面对角位有2种可能，无旋光异构体，3×2=6种

<Checkpoint>[1,2,5,6-共有6种](1)</Checkpoint>

（6）1,3,5,7-：2个旋光异构体，2种

<Checkpoint>[1,3,5,7-共有2种](1)</Checkpoint>

故共有70种

<Checkpoint>[共有70种同分异构体](1)</Checkpoint>