Circles in Triangles

The following pictures show n circles with the largest possible sum of radii packed inside a unit equilateral triangle. Apparently the cases n=1-10 and n=15 have been proven optimal.

1.
           2.
           3.
Σ r = √3/6 = .288+
Trivial. 		Σ r = 2√3/9 = .384+
Trivial. 		Σ r = (3√3 – 3) / 4 = .549+
Trivial.

4.
           5.
           6.
Σ r = .617+
Found by David W. Cantrell
in June 2011. 		Σ r = .693+
Found by David W. Cantrell
in June 2011. 		Σ r = 6 – 3√3 = .803+
Trivial.
 

7.
           8.
           9.
Σ r = .855+
Found by David W. Cantrell
in June 2011. 		Σ r = .912+
Found by David W. Cantrell
in June 2011. 		Σ r = .976+
Found by David W. Cantrell
in June 2011.

10.
           11.
           12.
Σ r = (15 – 5√3) / 6 = 1.056+
Trivial.
  		Σ r = 1.098+
Found by David W. Cantrell
in June 2011. 		Σ r = 1.143+
Found by David W. Cantrell
in June 2011.

13.
           14.
           15.
Σ r = 1.191+
Found by David W. Cantrell
in June 2011. 		Σ r = 1.243+
Found by David W. Cantrell
in June 2011. 		Σ r = (60 – 15√3) / 26 = 1.308+
Trivial.
 

16.
           17.
           18.
Σ r = 1.343+
Found by David W. Cantrell
in June 2011. 		Σ r = 1.381+
Found by David W. Cantrell
in June 2011. 		Σ r = 1.420+
Found by David W. Cantrell
in June 2011.

19.
           20.
           21.
Σ r = 1.462+
Found by David W. Cantrell
in June 2011. 		Σ r = 1.505+
Found by David W. Cantrell
in July 2011. 		Σ r = (105 – 21√3) / 44 = 1.559+
Trivial.
 

22.
           23.
           24.
Σ r = 1.589+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.622+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.655+
Found by David W. Cantrell
in July 2011.

25.
           26.
           27.
Σ r = 1.690+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.727+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.76421+
Found by Haowei Lin
in July 2026.

28.
           29.
           30.
Σ r = (168–28√3)/66 = 1.81064+
Trivial.
  		Σ r = 1.83673+
Found by Jonathan Viquerat
in June 2026. 		Σ r = 1.86580+
Found by Haowei Lin
in July 2026.

31.
           32.
           33.
Σ r = 1.89482+
Found by Joost de Winter
in October 2026. 		Σ r = 1.92402+
Found by Joost de Winter
in October 2026. 		Σ r = 1.95650+
Found by Joost de Winter
in October 2026.

34.
           35.
           36.
Σ r = 1.98837+
Found by Joost de Winter
in October 2026. 		Σ r = 2.02050+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 2.06137+
Found by Joost de Winter
in October 2026.

37.
           38.
           39.
Σ r = 2.08452+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 2.11045+
Found by Joost de Winter
in October 2026. 		Σ r = 2.13634+
Found by Joost de Winter
in October 2026.

40.
           41.
           42.
Σ r = 2.16253+
Found by Joost de Winter
in October 2026. 		Σ r = 2.19047+
Found by Joost de Winter
in October 2026. 		Σ r = 2.21804+
Found by Joost de Winter
in October 2026.

43.
           44.
           45.
Σ r = 2.24707+
Found by Joost de Winter
in October 2026. 		Σ r = 2.27566+
Found by Jonathan Viquerat
in October 2026. 		Σ r = 2.31194+
Found by Joost de Winter
in October 2026.

46.
           47.
           48.
Σ r = 2.33274+
Found by Joost de Winter
in October 2026. 		Σ r = 2.35619+
Found by Joost de Winter
in October 2026. 		Σ r = 2.37975+
Found by Joost de Winter
in October 2026.

49.
           50.
Σ r = 2.40336+
Found by Joost de Winter
in October 2026. 		Σ r = 2.42692+
Found by Joost de Winter
in October 2026.