Circles in Squares

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 = 1.866+
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.


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