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 = .796+ Found by David W. Cantrell in April 2011. |

4. | 5. | 6. | ||

Σ r = 1.006+ Found by David W. Cantrell in April 2011. | Σ r = (3 + √2)/4 = 1.866+ 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 = 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 = 2.634+ Found by David W. Cantrell in July 2011. | 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. | |

Σ r = 2.889+ Found by David W. Cantrell in July 2011. | Σ r = 2.936+ Found by Eckard Specht in January 2012. |