The following pictures show n unit squares packed inside the smallest known circle (of radius r).

Only the cases n=1 and n=2 have been proved optimal.

1. | 2. | 3. | ||

r = √2 / 2 = .707+ Trivial. | r = √5 / 2 = 1.118+ Trivial. | r = 5 √17 / 16 = 1.288+ Found by Erich Friedman in 1997. |

4. | 5. | 6. | ||

r = √2 = 1.414+ Found by Erich Friedman in 1997. | r = √10 / 2 = 1.581+ Found by Erich Friedman in 1997. | r = 1.688+ Found by Erich Friedman in 1997. |

7. | 8. | 9. | ||

r = √13 / 2 = 1.802+ Found by Erich Friedman in 1997. | r = 1.978+ Found by David W. Cantrell in March 2002. | r = √1105 / 16 = 2.077+ Found by Erich Friedman in 1997. |

10. | 11. | 12. | ||

r = 3 √2 / 2 = 2.121+ Found by Erich Friedman in 1997. | r = 2.214+ Found by David W. Cantrell in March 2002. | r = √5 = 2.236+ Found by Erich Friedman in 1997. |

13. | 14. | 15. | ||

r = 2.3607+ Found by David W. Cantrell in November 2010. | r = 2.5 Found by Erich Friedman in 1997. | r = 2.533+ Found by by David W. Cantrell in June 2002. |

16. | 17. | 18. | ||

r = √11009 / 40 = 2.623+ Found by Erich Friedman in 1997. | r = 2.677+ Found by Maurizio Morandi in December 2010. | r = √481 / 8 = 2.741+ Found by David W. Cantrell in March 2002. |

19. | 20. | 21. | ||

r = 2.807+ Found by David W. Cantrell in June 2002. | r = 2.893+ Found by David W. Cantrell in June 2002. | r = √34 / 2 = 2.915+ Found by Erich Friedman in 1997. |

22. | 23. | 24. | ||

r = 3.036+ Found by David W. Cantrell in June 2002. | r = 3.073+ Found by David W. Cantrell in June 2002. | r = 3.110+ Found by Maurizio Morandi in October 2010. |

25. | 26. | 27. | ||

r = √5809 / 24 = 3.175+ Found by David W. Cantrell in April 2002. | r = √41 / 2 = 3.201+ Found by Erich Friedman in 1997. | r = 13√145/48 = 3.261+ Found by Maurizio Morandi in September 2010. |

28. | 29. | 30. | ||

r = 3.3471+ Found by David W. Cantrell in November 2010. | r = 3.412+ Found by DC after MM in August 2012. | r = √27985 / 48 = 3.485+ Found by David W. Cantrell in March 2002. |

31. | 32. | 33. | ||

r = 5 √2 / 2 = 3.535+ Found by Erich Friedman in 1997. | r = √40385 / 56 = 3.588+ Found by Erich Friedman in 1997. | r = √13 = 3.605+ Found by Maurizio Morandi in June 2010. |

34. | 35. | |

r = 3.629+ Found by Maurizio Morandi in December 2010. | r = √53 / 2 = 3.640+ Found by David Eppstein in March 2002. |