(August 2013)

What if the centers need to be contained in the square, but the rest of the circle need not be? Again we ask for the smallest square that works.

n=1 s = 2 | n=2 s = 3 + √2 = 4.4142+ | n=3 s = 5 + 3/√2 = 7.1213+ |

n=4 s = 7 + 2 √2 = 9.8284+ | n=5 s = 12.5604+ | n=6 s = 15.4695+ |

n=7 s = 18.4994+ | n=8 s = 21.6882+ (Joe DeVincentis) | n=9 s = 25.3951+ (Maurizio Morandi) |

n=10 s = 28.6835+ (Joe DeVincentis) | n=11 s = 32.5383+ (Joe DeVincentis) | n=12 s = 36.4636+ (Joe DeVincentis) |

n=13 s = 40.3575+ (Joe DeVincentis) | n=14 s = 44.2073+ (Joe DeVincentis) | n=15 s = 48.2039+ (Joe DeVincentis) |

When only the centers of the circles need to be contained in the square of side s, the smallest-known sides are shown below:

n=2 s = √2 = 1.4142+ | n=3 s = 2 + 1/√2 = 2.7070+ | n=4 s = 3.8724+ |

n=5 s = 4.9836+ | n=6 s = 6.6691+ | n=7 s = 8.7329+ (Joe DeVincentis) |

n=8 s = 10.7169+ (Joe DeVincentis) | n=9 s = 13.0749+ (Joe DeVincentis) | n=10 s = 15.4216+ (Joe DeVincentis) |

n=11 s = 17.9409+ (Joe DeVincentis) | n=12 s = 20.2186+ (Joe DeVincentis) | n=13 s = 22.7292+ (Joe DeVincentis) |

n=14 s = 24.9614+ (Joe DeVincentis) | n=15 s = 27.6147+ (Joe DeVincentis) |

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/1/13.