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 = .500Trivial. Σ r = 2/3 = .666+Trivial. Σ r = 8(7 – 2√2) / 41 = .813+Found by David W. Cantrellin July 2011.

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

 7. 8. 9. Σ r = 1.324+Found by David W. Cantrellin July 2011. Σ r = 1.430+Found by David W. Cantrellin July 2011. Σ r = 1.528+Found by David W. Cantrellin July 2011.

 10. 11. 12. Σ r = 1.610+Found by David W. Cantrellin July 2011. Σ r = 1.685+Found by David W. Cantrellin July 2011. Σ r = 1.765+Found by David W. Cantrellin July 2011.

 13. 14. 15. Σ r = 1.835+Found by David W. Cantrellin July 2011. Σ r = 1.916+Found by David W. Cantrellin July 2011. Σ r = 1.985+Found by David W. Cantrellin July 2011.

 16. 17. 18. Σ r = 2.059+Found by David W. Cantrellin July 2011. Σ r = 2.117+Found by David W. Cantrellin July 2011. Σ r = 2.180+Found by David W. Cantrellin July 2011.

 19. 20. 21. Σ r = 2.241+Found by David W. Cantrellin July 2011. Σ r = 2.305+Found by David W. Cantrellin July 2011. Σ r = 2.364+Found by David W. Cantrellin July 2011.

 22. 23. 24. Σ r = 2.425+Found by David W. Cantrellin July 2011. Σ r = 2.484+Found by David W. Cantrellin July 2011. Σ r = 2.535+Found by David W. Cantrellin July 2011.

 25. 26. 27. Σ r = 2.592+Found by David W. Cantrellin July 2011. Σ r = 2.638+Found by David W. Cantrellin July 2011. Σ r = 2.687+Found by David W. Cantrellin July 2011.

 28. 29. 30. Σ r = 2.740+Found by David W. Cantrellin July 2011. Σ r = 2.792+Found by David W. Cantrellin July 2011. Σ r = 2.844+Found by David W. Cantrellin July 2011.

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