The following pictures show n circles with the largest possible sum of radii packed inside a unit square.
1.
2.
3.
Σ r = 1/2 = .500
Trivial.
Σ r = 2 – √2 = .585+
Trivial.
A continuum of solutions.
Σ r = .796+
Found by David W. Cantrell
in April 2011.
4.
5.
6.
Σ r = 1.006+
Found by David W. Cantrell
in April 2011.
Σ r = (3 + √2)/4 = 1.103+
Found by David W. Cantrell
in April 2011.
Σ r = 1.202+
Found by David W. Cantrell
in April 2011.
7.
8.
9.
Σ r = 1.306+
Found by David W. Cantrell
in April 2011.
Σ r = 1.423+
Found by David W. Cantrell
in April 2011.
Σ r = 1.524+
Found by David W. Cantrell
in April 2011.
10.
11.
12.
Σ r = 1.591+
Found by David W. Cantrell
in April 2011.
Σ r = 1.680+
Found by David W. Cantrell
in April 2011.
Σ r = 1.765+
Found by David W. Cantrell
in April 2011.
13.
14.
15.
Σ r = 1.829+
Found by David W. Cantrell
in April 2011.
Σ r = 1.905+
Found by David W. Cantrell
in April 2011.
Σ r = 1.980+
Found by David W. Cantrell
in April 2011.
16.
17.
18.
Σ r = 2.053+
Found by David W. Cantrell
in April 2011.
Σ r = 2.111+
Found by David W. Cantrell
in April 2011.
Σ r = 2.178+
Found by David W. Cantrell
in April 2011.
19.
20.
21.
Σ r = 2.236+
Found by David W. Cantrell
in April 2011.
Σ r = 2.301+
Found by David W. Cantrell
in July 2011.
Σ r = 2.362+
Found by David W. Cantrell
in July 2011.
22.
23.
24.
Σ r = 2.420+
Found by David W. Cantrell
in July 2011.
Σ r = 2.478+
Found by David W. Cantrell
in July 2011.
Σ r = 2.530+
Found by Eckard Specht
in January 2012.
25.
26.
27.
Σ r = 2.587+
Found by David W. Cantrell
in January 2012.
Σ r = 2.634+
Found by David W. Cantrell
in July 2011.
Σ r = 2.685+
Found by David W. Cantrell
in January 2012.
28.
29.
30.
Σ r = 2.737+
Found by David W. Cantrell
in July 2011.
Σ r = 2.790+
Found by David W. Cantrell
in July 2011.
Σ r = 2.842+
Found by David W. Cantrell
in July 2011.
31.
32.
Σ r = 2.889+
Found by David W. Cantrell
in July 2011.
Σ r = 2.936+
Found by Eckard Specht
in January 2012.