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

4. | 5. | 6. | ||

Σ r = 1.006+ Found by David W. Cantrell in July 2011. | Σ r = 1.112+ Found by David W. Cantrell in July 2011. | Σ r = 1.525+ Found by David W. Cantrell in July 2011. |

7. | 8. | 9. | ||

Σ r = 1.324+ Found by David W. Cantrell in July 2011. | Σ r = 1.430+ Found by David W. Cantrell in July 2011. | Σ r = 1.528+ Found by David W. Cantrell in July 2011. |

10. | 11. | 12. | ||

Σ r = 1.610+ Found by David W. Cantrell in July 2011. | Σ r = 1.685+ Found by David W. Cantrell in July 2011. | Σ r = 1.765+ Found by David W. Cantrell in July 2011. |

13. | 14. | 15. | ||

Σ r = 1.835+ Found by David W. Cantrell in July 2011. | Σ r = 1.916+ Found by David W. Cantrell in July 2011. | Σ r = 1.985+ Found by David W. Cantrell in July 2011. |

16. | 17. | 18. | ||

Σ r = 2.059+ Found by David W. Cantrell in July 2011. | Σ r = 2.117+ Found by David W. Cantrell in July 2011. | Σ r = 2.180+ Found by David W. Cantrell in July 2011. |

19. | 20. | 21. | ||

Σ r = 2.241+ Found by David W. Cantrell in July 2011. | Σ r = 2.305+ Found by David W. Cantrell in July 2011. | Σ r = 2.364+ Found by David W. Cantrell in July 2011. |

22. | 23. | 24. | ||

Σ r = 2.425+ Found by David W. Cantrell in July 2011. | Σ r = 2.484+ Found by David W. Cantrell in July 2011. | Σ r = 2.535+ Found by David W. Cantrell in July 2011. |

25. | 26. | 27. | ||

Σ r = 2.592+ Found by David W. Cantrell in July 2011. | Σ r = 2.638+ Found by David W. Cantrell in July 2011. | Σ r = 2.687+ Found by David W. Cantrell in July 2011. |

28. | 29. | 30. | ||

Σ r = 2.740+ Found by David W. Cantrell in July 2011. | Σ r = 2.792+ Found by David W. Cantrell in July 2011. | Σ r = 2.844+ Found by David W. Cantrell in July 2011. |