Packing 16, 17 or 18 circles in an equilateral triangle (Q1900192)

What is the smallest circle, or square, or equilateral triangle etc., that can contain \(n\) nonoverlapping unit circles? After giving a survey on results related to this problem, the authors consider the case of equilateral triangles for the special values \(n\in \{16, 17, 18\}\), filling a gap in the literature. They present efficient packings for these cases. Clearly, the problem of finding the largest common radius \(r_n\) of \(n\) congruent circles inside a unilateral triangle \(T\) is equivalent to maximizing the minimum pairwise distance \(t_n\) of \(n\) points in \(T\). By the homothety between \(T\) and the smallest equilateral triangle containing their centers, the relation \[ t_n= {{2r_n} \over {1-2 \sqrt {3r_n}}} \] between \(r_n\) and \(t_n\) is obtained. For the latter problem, two numerical methods generating near-optimal solutions are described.