= Solution
\b[TODO: numerical endpoints require the unreadable source diagram.] Let $\ell_{V,\max}$ and $\ell_{V,\min}$ be the maximum and minimum <main sequence> visual <luminosities>, in units of the <Sun>'s visual <luminosity>. At a distance $d$, the <inverse-square law> and the <magnitude difference> relation give
$$
m_V-m_{V,\odot}=-2.5\log_{10}\left[\ell_V\left(\frac{1\,\mathrm{AU}}d\right)^2\right].
$$
Thus, using $m_{V,\odot}=-26.77$,
$$
\boxed{d(\mathrm{pc})=\frac{10^{(m_V+26.77)/5}}{206264.806}\sqrt{\ell_V}.}
$$
At the naked-eye detection threshold $m_V=6$, the greatest distances for the intrinsically brightest and faintest <main sequence> stars would respectively be
$$
\boxed{d_{\max}=17.36\sqrt{\ell_{V,\max}}\,\mathrm{pc},\qquad d_{\min}=17.36\sqrt{\ell_{V,\min}}\,\mathrm{pc}.}
$$
The historical $m_V=1$ endpoint is a brightness reference, not the visibility threshold: if that endpoint were intended instead, the corresponding coefficient would be $1.736$ rather than $17.36$.
The source <Hertzsprung-Russell diagram> has lost the stellar loci and <luminosity> scale except for a surviving top tick; neither endpoint can be read reliably. If its scale was <bolometric luminosity> rather than visual <luminosity>, <bolometric corrections> would also be needed, especially for very hot and very cool stars. Equating the two <luminosity> ratios would then be an explicitly approximate assumption. These missing data are not replaced by invented endpoint values. <Interstellar extinction> is neglected in the displayed distance relation.
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