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

1. | 2. | 3. | ||

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

4. | 5. | 6. | ||

Σ r = 5 / 3 = 1.666+ Found by Veit Elser in November 2009. | Σ r = 1.866+ Found by Veit Elser in November 2009. | Σ r = 2.110+ Found by Veit Elser in November 2009. |

7. | 8. | 9. | ||

Σ r = 7 / 3 = 2.333+ Found by Veit Elser in November 2009. | Σ r = 2.512+ Found by Veit Elser in November 2009. | Σ r = 2.660+ Found by Veit Elser in November 2009. |

10. | 11. | 12. | ||

Σ r = 2.783+ Found by David W. Cantrell in April 2011. | Σ r = 2.947+ Found by Veit Elser in November 2009. | Σ r = 3.103+ Found by Veit Elser in November 2009. |

13. | 14. | 15. | ||

Σ r = 3.227+ Found by David W. Cantrell in April 2011. | Σ r = 3.359+ Found by Veit Elser in November 2009. | Σ r = 3.480+ Found by David W. Cantrell in April 2011. |

16. | 17. | 18. | ||

Σ r = 3.611+ Found by David W. Cantrell in April 2011. | Σ r = 3.730+ Found by David W. Cantrell in April 2011. | Σ r = 3.850+ Found by David W. Cantrell in April 2011. |

19. | 20. | 21. | ||

Σ r = 3.981+ Found by David W. Cantrell in May 2011. | Σ r = 4.076+ Found by David W. Cantrell in April 2011. | Σ r = 4.181+ Found by David W. Cantrell in June 2011. |

22. | 23. | 24. | ||

Σ r = 4.287+ Found by David W. Cantrell in June 2011. | Σ r = 4.372+ Found by David W. Cantrell in June 2011. | Σ r = 4.466+ Found by David W. Cantrell in June 2011. |

25. | 26. | 27. | ||

Σ r = 4.564+ Found by David W. Cantrell in June 2011. | Σ r = 4.664+ Found by David W. Cantrell in June 2011. | Σ r = 4.762+ Found by David W. Cantrell in June 2011. |

28. | 29. | 30. | ||

Σ r = 4.849+ Found by David W. Cantrell in June 2011. | Σ r = 4.934+ Found by David W. Cantrell in June 2011. | Σ r = 5.019+ Found by David W. Cantrell in June 2011. |

31. | 32. | |

Σ r = 5.104+ Found by David W. Cantrell in June 2011. | Σ r = 5.191+ Found by David W. Cantrell in June 2011. |