  In this work, we consider a number of problems defined on the triangular
lattice with $n$ rows, which we will denote as $T_n$. Define a \textit{proper
coloring} to be an assignment of colors to the points of $T_n$ such that no
three points constituting the vertices of an equilateral triangle all receive
the same color, and denote by $f(n)$ the smallest possible number of colors
that can be used in a proper coloring of $T_n$. We either determine exactly or
give upper bounds for $f(n)$ for many small values of $n$, and it is shown that
$\lim_{n\to\infty} \frac{f(n)}{n} \leq \frac13$. We also give formulas counting
the number of pairs of points in $T_n$ for which there are, respectively, 0, 1,
or 2 choices of points in $T_n$ which extend those two into the vertices of an
equilateral triangle. Along the way, we pose a number of related questions.
