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 = 4 – 3/√2 – √(4–2√2) = .796+
Found by David W. Cantrell
in April 2011.

4.
           5.
           6.
Σ r = 6 – 2√2 – 2√(4–2√2) = 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 = 6√2 – 4 – 4√(2–√2) = 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 = 2.63598+
Found by Haowei Lin
in July 2026. 		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.
           33.
Σ r = 2.889+
Found by David W. Cantrell
in July 2011. 		Σ r = 2.939+
Found by Timo Berthold et al
in January 2026. 		Σ r = 2.98728+
Found by Haowei Lin
in July 2026.

34.
           35.
           36.
Σ r = 3.02979+
Found by Haowei Lin
in July 2026. 		Σ r = 3.07403+
Found by Haowei Lin
in July 2026. 		Σ r = 3.12175+
Found by Haowei Lin
in July 2026.

37.
           38.
           39.
Σ r = 3.16149+
Found by Haowei Lin
in July 2026. 		Σ r = 3.20594+
Found by Haowei Lin
in July 2026. 		Σ r = 3.24811+
Found by Haowei Lin
in July 2026.

40.
Σ r = 3.29239+
Found by Haowei Lin
in July 2026.