Circles in Rectangles

The following pictures show n circles with the largest possible sum of radii packed inside a rectangle of perimeter 4.

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 = 1.006+
Found by David W. Cantrell
in July 2011. Σ r = 1.112+
Found by David W. Cantrell
in July 2011. Σ r = 1.525+
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 = 2.364+
Found by David W. Cantrell
in July 2011.

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 = 2.638+
Found by David W. Cantrell
in July 2011. Σ r = 2.687+
Found by David W. Cantrell
in July 2011.

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.

4.		5.		6.
Σ r = 1.006+ Found by David W. Cantrell in July 2011.		Σ r = 1.112+ Found by David W. Cantrell in July 2011.		Σ r = 1.525+ 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 = 2.364+ Found by David W. Cantrell in July 2011.

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 = 2.638+ Found by David W. Cantrell in July 2011.		Σ r = 2.687+ Found by David W. Cantrell in July 2011.

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.