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 = .500Trivial. Σ r = 2 – √2 = .585+Trivial.A continuum of solutions. Σ r = .796+Found by David W. Cantrellin April 2011.

 4. 5. 6. Σ r = 1.006+Found by David W. Cantrellin April 2011. Σ r = (3 + √2)/4 = 1.866+Found by David W. Cantrellin April 2011. Σ r = 1.202+Found by David W. Cantrellin April 2011.

 7. 8. 9. Σ r = 1.306+Found by David W. Cantrellin April 2011. Σ r = 1.423+Found by David W. Cantrellin April 2011. Σ r = 1.524+Found by David W. Cantrellin April 2011.

 10. 11. 12. Σ r = 1.591+Found by David W. Cantrellin April 2011. Σ r = 1.680+Found by David W. Cantrellin April 2011. Σ r = 1.765+Found by David W. Cantrellin April 2011.

 13. 14. 15. Σ r = 1.829+Found by David W. Cantrellin April 2011. Σ r = 1.905+Found by David W. Cantrellin April 2011. Σ r = 1.980+Found by David W. Cantrellin April 2011.

 16. 17. 18. Σ r = 2.053+Found by David W. Cantrellin April 2011. Σ r = 2.111+Found by David W. Cantrellin April 2011. Σ r = 2.178+Found by David W. Cantrellin April 2011.

 19. 20. 21. Σ r = 2.236+Found by David W. Cantrellin April 2011. Σ r = 2.301+Found by David W. Cantrellin July 2011. Σ r = 2.362+Found by David W. Cantrellin July 2011.

 22. 23. 24. Σ r = 2.420+Found by David W. Cantrellin July 2011. Σ r = 2.478+Found by David W. Cantrellin July 2011. Σ r = 2.530+Found by Eckard Spechtin January 2012.

 25. 26. 27. Σ r = 2.587+Found by David W. Cantrellin January 2012. Σ r = 2.634+Found by David W. Cantrellin July 2011. Σ r = 2.685+Found by David W. Cantrellin January 2012.

 28. 29. 30. Σ r = 2.737+Found by David W. Cantrellin July 2011. Σ r = 2.790+Found by David W. Cantrellin July 2011. Σ r = 2.842+Found by David W. Cantrellin July 2011.

 31. 32. Σ r = 2.889+Found by David W. Cantrellin July 2011. Σ r = 2.936+Found by Eckard Spechtin January 2012.

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