"CageWalking”Strategy is a strategy applied to metal-catalyzed functionalization of carboranes. The "cage walking" phenomenon was first observed by Hawthorne in 1971 in the nido-$\mathrm { C _ { 2 } B _ { 9 } H _ { 1 2 } }$ system, which means that the metal unit on the carborane cage undergoes 1,2-migration. The charge density on the $\rm{C}$ or $\rm{B}$ atoms of the carborane cage is one of the important factors controlling the migration of metal atoms. The structures of $o-\mathrm { C _2B_{10}H_{1 2} }$ and $m-\mathrm { C _2B_{10}H_{1 2} }$ are shown in the following figure: black represents $\rm{CH_2}$, white represents $\rm{BH}$

$o-\mathrm { C _2B_{10}H_{1 2} }$:

<MultiModal>![[H]C1234B567([H])C189([H])B2%10%11([H])B3%12%13([H])B54%14([H])B6%15%16([H])B87%17([H])B9%10%18([H])B%17%15%19([H])B%14%12%16([H])B%18%11%13%19[H]](/media/uploads/6cbc7f60-c4e8-4c39-be7f-6e1bc2791bb5.png)</MultiModal>

$m-\mathrm { C _2B_{10}H_{1 2} }$:

<MultiModal>![[H]B1234C567([H])B189([H])C2%10%11([H])B3%12%13([H])B54%14([H])B6%15%16([H])B87%17([H])B9%10%18([H])B%17%15%19([H])B%14%12%16([H])B%18%11%13%19[H]](/media/uploads/a30e19b7-6653-43e0-9066-f672ade591a7.png)</MultiModal>

Define the distance between two vertices on the icosahedral framework as the minimum number of edges required to connect the two vertices. Thus, the chemical environment of a boron atom can be determined based on its distance to the two carbon atoms. Use an ordered pair $(a,b)$ to represent the chemical environment of a boron atom, where $a$ is the smaller of the distances from the boron atom to the two carbon atoms,
$b$ is the larger of the distances from the boron atom to the two carbon atoms, and $(a,b)$ is called the distance pair of the boron atom.

Sort the charge densities of all $\rm{B}$ atoms in the two carboranes from high to low, respectively (boron atoms with the same distance pair are recorded only once), and map the two groups of sorted distance pairs to points in the Cartesian coordinate system, respectively. Connect the points in order to obtain two sets of broken lines. The total length of the broken line obtained by $o-\mathrm { C _2B_{10}H_{1 2} }$ is denoted as $x$, and the total length of the broken line obtained by $m-\mathrm { C _2B_{10}H_{1 2} }$ is denoted as $y$. Then the value of $\frac{x+y}{x-y}$ is:
“笼行走”策略("CageWalking”Strategy)是一种应用于碳硼烷的金属催化官能团化策略。“笼行走”现象最早由Hawthorne于1971年在nido-$\mathrm {  C _ { 2 } B _ { 9 } H _ { 1 2 } }$ 体系中观察到，其含义是碳硼烷笼上的金属单元发生1,2-迁移。碳硼烷笼上$\rm{C}$原子或$\rm{B}$原子上的电荷密度是控制金属原子迁移的重要因素之一。$o-\mathrm { C _2B_{10}H_{1 2} }$ 和 $m-\mathrm { C _2B_{10}H_{1 2} }$ 的结构已展示于下图中：黑色代表$\rm{CH_2}$，白色代表$\rm{BH}$

$o-\mathrm { C _2B_{10}H_{1 2} }$：

<MultiModal>![[H]C1234B567([H])C189([H])B2%10%11([H])B3%12%13([H])B54%14([H])B6%15%16([H])B87%17([H])B9%10%18([H])B%17%15%19([H])B%14%12%16([H])B%18%11%13%19[H]](/media/uploads/6cbc7f60-c4e8-4c39-be7f-6e1bc2791bb5.png)</MultiModal>

$m-\mathrm { C _2B_{10}H_{1 2} }$：

<MultiModal>![[H]B1234C567([H])B189([H])C2%10%11([H])B3%12%13([H])B54%14([H])B6%15%16([H])B87%17([H])B9%10%18([H])B%17%15%19([H])B%14%12%16([H])B%18%11%13%19[H]](/media/uploads/a30e19b7-6653-43e0-9066-f672ade591a7.png)</MultiModal>

定义正二十面体框架上两个顶点的距离为连接两个顶点所需要边的数目的最小值。由此可以根据到两个碳原子的距离来确定硼原子的化学环境。用一个有序数对$(a,b)$来表示硼原子的化学环境，其中$a$为硼原子到两个碳原子距离中较小的那个，
$b$为硼原子到两个碳原子距离中较大的那个，$(a,b)$称为该硼原子的距离数对。

分别将两个碳硼烷中所有$\rm{B}$原子的电荷密度从高到低排序（具有相同距离数对的硼原子只记录一次），分别将两组排序好的距离数对对应到平面直角坐标系的点，按顺序连接点，得到两组折线，$o-\mathrm { C _2B_{10}H_{1 2} }$所得到的折线总长度记作$x$，$m-\mathrm { C _2B_{10}H_{1 2} }$所得到的折线总长度记作$y$,则$\frac{x+y}{x-y}$的值为：
All other options are incorrect
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$\rm{C}$'s electronegativity is greater than $\rm{B}$'s, mainly considering $\rm{C}$'s inductive electron-withdrawing effect.

<Checkpoint>[Considering $\rm{C}$'s inductive electron-withdrawing effect](1)</Checkpoint>

Therefore, the farther away from the $\rm{C}$ atom, the greater the charge density of the $\rm{B}$ atom;

<Checkpoint>[The farther away from the $\rm{C}$ atom, the greater the charge density of the $\rm{B}$ atom](1)</Checkpoint>

Based on this, the charge density of the $\rm{B}$ atoms in $o-\mathrm { C _2B_{10}H_{1 2} }$ can be ranked from high to low:

$(2,3)>(2,2)>(1,2)>(1,1)$

Therefore, it can be determined that $x=3$.

<Checkpoint>[$x=3$](2)</Checkpoint>

Since the attenuation of the inductive electron-withdrawing effect is relatively severe, the charge density of $(1,3)$, which is directly connected to the $\rm{C}$ atom, is less than that of $(2,2)$, which is not directly connected to the $\rm{C}$ atom.

<Checkpoint>[The attenuation of the inductive electron-withdrawing effect is relatively severe](1)</Checkpoint>

<Checkpoint>[The charge density of $(1,3)$, which is directly connected to the $\rm{C}$ atom, is less than that of $(2,2)$, which is not directly connected to the $\rm{C}$ atom](1)</Checkpoint>

Therefore, the charge density of the $\rm{B}$ atoms in $m-\mathrm { C _2B_{10}H_{1 2} }$ can be ranked from high to low:

$(2,2)>(1,3)>(1,2)>(1,1)$

Therefore, it can be determined that $y=2+\sqrt{2}$.

<Checkpoint>[$y=2+\sqrt{2}$](2)</Checkpoint>

$\frac{x+y}{x-y}=\frac{\sqrt{2}+5}{1-\sqrt{2}}\approx-15.49$

<Checkpoint>[$\frac{x+y}{x-y}\approx-15.49$](1)</Checkpoint>
$\rm{C}$的电负性比$\rm{B}$大，主要考虑$\rm{C}$的诱导吸电子作用

<Checkpoint>[考虑$\rm{C}$的诱导吸电子作用](1)</Checkpoint>

因此离$\rm{C}$原子越远的$\rm{B}$原子电荷密度越大；

<Checkpoint>[离$\rm{C}$原子越远的$\rm{B}$原子电荷密度越大](1)</Checkpoint>

据此可以得到$o-\mathrm { C _2B_{10}H_{1 2} }$中$\rm{B}$原子的电荷密度从高到低排序:

$(2,3)>(2,2)>(1,2)>(1,1)$

因此可判断$x=3$。

<Checkpoint>[$x=3$](2)</Checkpoint>

由于诱导吸电子作用的衰减较为严重，因此与$\rm{C}$原子直接相连的$(1.3)$电荷密度小于与$\rm{C}$原子不直接相连的$(2,2)$。

<Checkpoint>[诱导吸电子作用的衰减较为严重](1)</Checkpoint>

<Checkpoint>[与$\rm{C}$原子直接相连的$(1.3)$电荷密度小于与$\rm{C}$原子不直接相连的$(2,2)$](1)</Checkpoint>

因此可以得到$m-\mathrm { C _2B_{10}H_{1 2} }$中$\rm{B}$原子的电荷密度从高到低排序：

$(2,2)>(1,3)>(1,2)>(1,1)$

因此可判断$y=2+\sqrt{2}$。

<Checkpoint>[$y=2+\sqrt{2}$](2)</Checkpoint>

$\frac{x+y}{x-y}=\frac{\sqrt{2}+5}{1-\sqrt{2}}\approx-15.49$

<Checkpoint>[$\frac{x+y}{x-y}\approx-15.49$](1)</Checkpoint>