When k = 2, this becomes the problem of packing of squares in a loop. What are the smallest rectangles you can find for various n? When k ≥ 3, we need the squares to form the vertices of a k-regular graph. Can you find small packings for k=3? Can you find any packings for k=4 or k=5? Can you see why packings are impossible for k ≥ 6?

Claudio Baiocchi noted that kn must be even to form a n-vertex k-regular graph.

Claudio Baiocchi also noted that when k=3, an n=4 solution and an n=6 solution exist when corner contacts are allowed.

If we relax the condition that the squares form a connected group, there are solutions for k=0 and k=1 as well. The k=0 solutions are related to the solutions with no connectivity constraints.

Below are the smallest known packings:

n=2 area=6 | n=4 area=40 | n=6 area=108 | n=8 area=240 | n=10 area=435 | n=12 area=735 (Maurizio Morandi) |

n=14 area=1125 | n=16 area=1647 (Maurizio Morandi) |

n=18 area=2294 (Maurizio Morandi) | n=20 area=3102 (Maurizio Morandi) |

n=3 area=15 | n=4 area=35 (Maurizio Morandi) | n=5 area=63 | n=6 area=99 | n=7 area=154 (Maurizio Morandi) | n=8 area=224 (Claudio Baiocchi) | n=9 area=315 |

n=10 area=425 | n=11 area=546 | n=12 area=690 (Maurizio Morandi) |

n=13 area=874 (Maurizio Morandi) | n=14 area=1081 (Maurizio Morandi) |

n=15 area=1320 (Maurizio Morandi) | n=16 area=1590 (Maurizio Morandi) |

n=17 area=1890 (Maurizio Morandi) | n=18 area=2220 (Maurizio Morandi) |

n=8 area=221 | n=10 area=425 | n=12 area=693 |

n=12 area=899 (Andrew Bayly) | n=14 area=1295 (Maurizio Morandi) | n=15 area=1496 (Andrew Bayly) |

n=16 area=2009 | n=17 area=2220 (Maurizio Morandi) |

n=18 area=2627 | n=24 area=6125 (Andrew Bayly) |

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/14/13.