Circles in Triangles

The following pictures show n unit circles packed inside the smallest equilateral triangle (of side length s ). Most of these have been proved optimal. When m = (n2+n)/2, s = 2(n-1) + 23. It is conjectured that one less circle does not change this.

1.            2.            3.
s = 23 = 3.464+
Trivial. 		s = 2 + 23 = 5.464+
Trivial. 		s = 2 + 23 = 5.464+
Trivial.

4.            5.            6.
s = 43 = 6.928+
Proved by Milano in 1987. 		s = 4 + 23 = 7.464+
Proved by Milano in 1987. 		s = 4 + 23 = 7.464+
Proved by Oler/Groemer in 1961.

7.            8.            9.
s = 2 + 43 = 8.928+
Proved by Melissen in 1993. 		s = 9.29+
Proved by Melissen in 1993. 		s = 6 + 23 = 9.464+
Proved by Melissen in 1993.

10.            11.            12.
s = 6 + 23 = 9.464+
Proved by Oler/Groemer in 1961. 		s =10.73+
Proved by Melissen in 1993. 		s = 4 + 43 = 10.928+
Proved by Melissen in 1994.

13.            14.            15.
s = 11.40+
Found by Melissen in 1993. 		s = 8 + 23 = 11.464+
Found by Erdos/Oler in 1961. 		s = 8 + 23 = 11.464+
Proved by Erdos/Groemer in 1961.

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