For $c>0$, let $X_c$ denote the set of $x\in\mathbb{ R}\backslash\mathbb{ Q}$ such that $\left| x-\frac{ p}{ q} \right|<\frac{ 1}{ cq^2}$ has only finitely many rational solutions $\frac{ p}{ q}$. It is a classical fact, known since the 1950s, that $X_c$ is uncountable for $c>3$ and countable for $c<3$. However, the cardinality of $X_3$ does not appear to be present in the literature. We prove that $X_3$ is uncountable. More generally, we show that for any $n\in\mathbb{ N}\cup\{ \infty\}$, the set of $x\in\mathbb{ R}\backslash\mathbb{ Q}$ with Lagrange value exactly $3$ and such that $\left| x-\frac{ p}{ q} \right|<\frac{ 1}{ 3q^2}$ has exactly $n$ rational solutions $\frac{ p}{ q}$ is also uncountable.
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