Circles in Circles

The following pictures show n circles with the largest possible sum of radii packed inside a unit circle.

 1. 2. 3. Σ r = 1Trivial. Σ r = 1Trivial.A continuum of solutions. Σ r = 6√3 – 9 = 1.392+Trivial.

 4. 5. 6. Σ r = 5 / 3 = 1.666+Found by Veit Elserin November 2009. Σ r = 1.866+Found by Veit Elserin November 2009. Σ r = 2.110+Found by Veit Elserin November 2009.

 7. 8. 9. Σ r = 7 / 3 = 2.333+Found by Veit Elserin November 2009. Σ r = 2.512+Found by Veit Elserin November 2009. Σ r = 2.660+Found by Veit Elserin November 2009.

 10. 11. 12. Σ r = 2.783+Found by David W. Cantrellin April 2011. Σ r = 2.947+Found by Veit Elserin November 2009. Σ r = 3.103+Found by Veit Elserin November 2009.

 13. 14. 15. Σ r = 3.227+Found by David W. Cantrellin April 2011. Σ r = 3.359+Found by Veit Elserin November 2009. Σ r = 3.480+Found by David W. Cantrellin April 2011.

 16. 17. 18. Σ r = 3.611+Found by David W. Cantrellin April 2011. Σ r = 3.730+Found by David W. Cantrellin April 2011. Σ r = 3.850+Found by David W. Cantrellin April 2011.

 19. 20. 21. Σ r = 3.981+Found by David W. Cantrellin May 2011. Σ r = 4.076+Found by David W. Cantrellin April 2011. Σ r = 4.181+Found by David W. Cantrellin June 2011.

 22. 23. 24. Σ r = 4.287+Found by David W. Cantrellin June 2011. Σ r = 4.372+Found by David W. Cantrellin June 2011. Σ r = 4.466+Found by David W. Cantrellin June 2011.

 25. 26. 27. Σ r = 4.564+Found by David W. Cantrellin June 2011. Σ r = 4.664+Found by David W. Cantrellin June 2011. Σ r = 4.762+Found by David W. Cantrellin June 2011.

 28. 29. 30. Σ r = 4.849+Found by David W. Cantrellin June 2011. Σ r = 4.934+Found by David W. Cantrellin June 2011. Σ r = 5.019+Found by David W. Cantrellin June 2011.

 31. 32. Σ r = 5.104+Found by David W. Cantrellin June 2011. Σ r = 5.191+Found by David W. Cantrellin June 2011.

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