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.[1][2][3]

A conjecture of Paul Erdős and Norman Oler states that, if n is a triangular number, then the optimal packings of n − 1 and of n circles have the same side length: that is, according to the conjecture, an optimal packing for n − 1 circles can be found by removing any single circle from the optimal hexagonal packing of n circles.[4] This conjecture is now known to be true for n ≤ 15.[5]

Minimum solutions for the side length of the triangle:[1]

Number of circles Length
1 3.464...
2 5.464...
3 5.464...
4 6.928...
5 7.464...
6 7.464...
7 8.928...
8 9.293...
9 9.464...
10 9.464...
11 10.730...
12 10.928...
13 11.406...
14 11.464...
15 11.464...

A closely related problem is to cover the equilateral triangle with a given number of circles, having as small a radius as possible.[6]