= Solution
\b[$\mathrm{A=K[B_5H_8]}$ is <nido>; $\mathrm{B=K[B_6H_{11}]}$ and $\mathrm{C=B_6H_{12}}$ are <arachno>.] The initial <hydride> removes an acidic <bridging hydrogen> as a <proton>, releasing $\mathrm{H_2}$ and retaining the <potassium> counterion. Addition of one $\mathrm{BH_3}$ unit expands the <boron> framework; final protonation supplies the neutral hexaborane and KCl. These assignments conserve all the atoms in the displayed sequence, without inserting an unshown dehydrogenation step.
For $\mathrm{B_nH_m^{q-}}$, take one terminal B–H bond per vertex and remove its $2n$ electrons from the valence total. The <skeletal electron count> is $3n+m+q-2n=n+m+q$. Thus
$$
\begin{array}{c|c|c|c}
\text{cluster}&n&\text{skeletal electrons}&\text{skeletal pairs}\\\hline
[\mathrm{B_5H_8}]^-&5&5+8+1=14&7=n+2\\
[\mathrm{B_6H_{11}}]^-&6&6+11+1=18&9=n+3\\
\mathrm{B_6H_{12}}&6&6+12=18&9=n+3
\end{array}
$$
<Wade's rules> assign $n+1$, $n+2$ and $n+3$ pairs to <closo>, <nido> and <arachno> frameworks respectively. A derives from a six-vertex <closo> octahedron by removal of one vertex, giving a square-pyramidal <boron> skeleton. B and C have six-vertex open frameworks formally derived from an eight-vertex <closo> polyhedron by removing two vertices. Their additional hydrogens are distributed over bridging and other open-face positions; counting every <hydrogen> as an independent terminal B–H vertex would give the wrong classification.
The <arachno> assignment of $[\mathrm{B_6H_{11}}]^-$ has specific theoretical and NMR support. It should not be replaced by a permanently localized, unrearranged $[\mathrm{B_5H_8}]^-\!\cdot\!\mathrm{BH_3}$ picture: the corresponding symmetric adduct participates in fluxional rearrangement rather than being the lowest-energy structure.<The Structure of arachno-B6H11-, at -25 degrees C in (CD3)2O, Is Resolved via the ab Initio IGLO-GIAO NMR Procedure by Matthias Hofmann and others>[2000, abstract: arachno minimum, bridging-hydrogen removal from B6H12 and alternative-adduct comparison]{ref}
\Image[/past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1-borane-counts.png]
{title=Original skeletal-electron accounting for the nido-to-arachno borane sequence}
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