Circles in Rectangles

The following pictures show n circles with the largest possible sum of radii packed inside a rectangle of perimeter 4. Apparently the cases n=1-9 have been proved optimal.

1.
           2.
           3.
Σ r = 1/2 = .500
Trivial. 		Σ r = 2/3 = .666+
Trivial. 		Σ r = 8(7 – 2√2) / 41 = .813+
Found by David W. Cantrell
in July 2011.

4.
           5.
           6.
Σ r = 6–2√2–2√(4–2√2) = 1.006+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.112+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.212+
Found by David W. Cantrell
in July 2011.

7.
           8.
           9.
Σ r = 1.324+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.430+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.528+
Found by David W. Cantrell
in July 2011.

10.
           11.
           12.
Σ r = 1.610+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.685+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.765+
Found by David W. Cantrell
in July 2011.

13.
           14.
           15.
Σ r = 1.835+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.916+
Found by David W. Cantrell
in July 2011. 		Σ r = 1.985+
Found by David W. Cantrell
in July 2011.

16.
           17.
           18.
Σ r = 2.059+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.117+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.180+
Found by David W. Cantrell
in July 2011.

19.
           20.
           21.
Σ r = 2.241+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.305+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.365+
Found by Timo Berthold et al
in January 2026.

22.
           23.
           24.
Σ r = 2.425+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.484+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.535+
Found by David W. Cantrell
in July 2011.

25.
           26.
           27.
Σ r = 2.592+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.63932+
Found by Shaoheng Lai
in August 2026. 		Σ r = 2.69152+
Found by Everett Dutton
in August 2026.

28.
           29.
           30.
Σ r = 2.740+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.792+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.844+
Found by David W. Cantrell
in July 2011.

31.
           32.
           33.
Σ r = 2.89238+
Found by Aaden Abraham
in October 2026. 		Σ r = 2.94146+
Found by Jean-René Denoual
in October 2026. 		Σ r = 2.99118+
Found by Jean-René Denoual
in October 2026.

34.
           35.
           36.
Σ r = 3.03397+
Found by Jean-René Denoual
in October 2026. 		Σ r = 3.07913+
Found by Jean-René Denoual
in October 2026. 		Σ r = 3.12681+
Found by Jean-René Denoual
in October 2026.

37.
           38.
           39.
Σ r = 3.16593+
Found by Jean-René Denoual
in October 2026. 		Σ r = 3.20723+
Found by Jean-René Denoual
in October 2026. 		Σ r = 3.25236+
Found by Joost de Winter
in October 2026.

40.
           41.
           42.
Σ r = 3.29690+
Found by Jean-René Denoual
in October 2026. 		Σ r = 3.34065+
Found by Aaden Abraham
in October 2026. 		Σ r = 3.38312+
Found by Joost de Winter
in October 2026.

43.
           44.
           45.
Σ r = 3.42598+
Found by Jean-René Denoual
in October 2026. 		Σ r = 3.46697+
Found by Jean-René Denoual
in October 2026. 		Σ r = 3.50737+
Found by Jean-René Denoual
in October 2026.

46.
           47.
           48.
Σ r = 3.54814+
Found by Aaden Abraham
in October 2026. 		Σ r = 3.58361+
Found by Jean-René Denoual
in October 2026. 		Σ r = 3.62321+
Found by Jean-René Denoual
in October 2026.

49.
           50.
Σ r = 3.66158+
Found by Jean-René Denoual
in October 2026. 		Σ r = 3.69531+
Found by Jean-René Denoual
in October 2026.