Circle packing in an equilateral triangle

Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack $n$ unit circles into the smallest possible equilateral triangle. Optimal solutions are known for $n < 13$ and for any triangular number of circles, and conjectures are available for $n < 28$.

A conjecture of Paul Erdős and Norman Oler states that, if $n$ is a triangular number, then the optimal packings of $n &minus; 1$ and of $n$ circles have the same side length: that is, according to the conjecture, an optimal packing for $n &minus; 1$ circles can be found by removing any single circle from the optimal hexagonal packing of $n$ circles. This conjecture is now known to be true for $n ≤ 15$.

Minimum solutions for the side length of the triangle:

A closely related problem is to cover the equilateral triangle with a fixed number of equal circles, having as small a radius as possible.