{4} | {5} | {6} |

Call a collection of squares **k-balanced** if it has an equal number k of squares of each size. If we have a finite k-balanced neighborly collection of squares, what is the smallest possible value of k?

Given a rectangle, how can it be tiled with squares to maximize the proportion of the squares that are neighborly? What proportion of squares an infinite strip of a fixed width can be neighborly?

The known solutions are shown below.

{5} | {6} | {4,6} | {5,6} | {4,7} | {5,7} | {4,6,7} (MM) |

{5,6,7} (JD) | {5,8} | {4,5,8} (JD) | {5,6,8} (MM) | {4,5,6,8} | {4,9} (JD) | {4,10} (JD) |

{4,8,10} (MM) | {4,11} (JD) | {4,5,12} (MM) | {4,5,6,8,12} (MM) | {4,5,8,12,16} (MM) |

{4,5,8,12,20} (MM) | {4,5,6,8,12,20} (MM) | {4,5,6,8,12,20,28} (MM) |

The tilings below are infinite families, because blocks of 4's can be replaced by larger multiples of 4. Furthermore, these can be combined.

{4} | {4,6,8} (MM) | {4,5,7,8} (MM) | {4,5,8,10} (MM) | {4,5,10,16} (MM) | |

{4,5,6,8,12,16} (MM) | {4,5,8,12,16,20} (MM) | ||||

{4,5,6,8,12,20,24} (MM) | {4,5,6,8,12,36} (MM) | {4,5,6,8,16,36} (MM) | |||

{4,5,6,8,32,36} (MM) | {4,5,6,8,12,40} (MM) | {4,5,6,8,16,40} (MM) | |||

{4,5,6,8,32,48} (MM) | {4,5,6,12+4n,44+4n+4m} (MM) |

Joe DeVincentis showed that {4,N} neighborly tilings always exist for N≥6, by generalizing the {4,8}, {4,9}, {4,10}, and {4,11} tilings. Maurizio Morandi showed that {4,4n,4n+8+m} tilings exist for n>1 and m≥0. Joe DeVincentis also proved that a {4,5} planar tiling is not possible.

{2} | {3} | {4} | {3,6} | {3,4,6} |

The strips below were found by Maurizio Morandi. The ones in the last column can be combined with the plane tilings above.

{1} | {2} | {1,2} | {3} |

{1,3} | {2,3} | {1,2,3} (GS) | {4} |

{1,4} | {2,4} | {1,2,4} | {3,4} |

{1,3,4} | {2,3,4} | {1,2,3,4} | {5} |

{1,5} | {2,5} | {1,2,5} | {3,5} (MM) |

{1,3,5} | {2,3,5} | {1,2,3,5} | {4,5} |

{1,4,5} | {2,4,5} | {1,2,4,5} | {3,4,5} (JD) |

{1,3,4,5} (MM) | {2,3,4,5} | {1,2,3,4,5} | {6} |

{1,6} (MM) | {2,6} (MM) | {1,2,6} | {3,6} (MM) |

{1,3,6} | {2,3,6} | {1,2,3,6} | {4,6} |

{1,4,6} (MM) | {2,4,6} (MM) | {1,2,4,6} | {3,4,6} (MM) |

{1,3,4,6} (MM) | {2,3,4,6} (MM) | {1,2,3,4,6} | {5,6} |

{1,5,6} (MM) | {2,5,6} (MM) | {1,2,5,6} | possible with hundreds of squares {3,5,6} |

{1,3,5,6} (MM) | {2,3,5,6} | {1,2,3,5,6} (MM) | {4,5,6} |

{1,4,5,6} (MM) | {2,4,5,6} (MM) | {1,2,4,5,6} (MM) | ? {3,4,5,6} |

{1,3,4,5,6} (MM) | {2,3,4,5,6} (MM) | {1,2,3,4,5,6} (MM) | {7} |

Tilings of Rectangles Known

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | |||||||||||

2 | 1 | 0 | ||||||||||

3 | 2/3 | 1/3 | 0 | |||||||||

4 | 1/2 | 1/5 | 1/4 | 1 | ||||||||

5 | 2/5 | 1/2 | 1/7 | 1/4 | 1/8 | |||||||

6 | 1/3 | 1/3 | 1/7 | 2/3 | 4/5 | 2/3 (GA) | ||||||

7 | 2/7 | 1/4 | 2/5 | 2/7 | 2/7 | 3/5 (MM) | 4/9 | |||||

8 | 1/4 | 1/2 | 1/4 | 1 | 1/3 | 2/3 (GS) | 1/3 (GS) | 1/2 (GA) | ||||

9 | 2/9 | 1/3 | 2/11 | 3/7 (MM) | 4/9 | 1/2 (MM) | 3/7 (GS) | 3/7 (GS) | 5/11 | |||

10 | 1/5 | 3/5 | 1/4 | 5/7 | 1/3 (GS) | 5/7 | 2/5 (GS) | 1/2 (MM) | 4/9 (GS) | 6/11 | ||

11 | 2/11 | 3/7 | 2/9 | 4/9 (GS) | 1/3 (MM) | 1/2 | 1/2 (GS) | 5/11 (GS) | 2/5 (GS) | 4/9 (GS) | 5/14 (GS) | |

12 | 1/6 | 2/3 | 2/9 (MM) | 2/3 | 2/5 | 1/2 | 3/5 (GS) | 2/3 (GA) | 5/9 (GA) | 4/7 (GS) | 1/2 (GS) | 8/13 |

1 0 | 2 1 | 3 2/5 | 4 1/3 (MM) | 5 2/9 | 6 1 | 7 5/7 | 8 1 | 9 1 |

10 3/4+ε (MM) | 11 5/7 (MM) | 12 1 |

13 17/21 (MM) | 14 20/27 (MM) | 15 23/25 (MM) | 16 19/23 (MM) |

17 4/5 (MM) | 18 1 | 19 1 | 20 13/15 (MM) |

21 22/23 (MM) |

22 14/15 (MM) | 23 17/18 (MM) | 24 1 (MM) |

1 0 | 2 3/4 | 3 1/2 | 4 3/7 | 5 3/10 | 6 1/3 (MM) | 7 1/4 (MM) | 8 6/13 | 9 4/11 (MM) | 10 7/16 (MM) |

11 8/19 (MM) | 12 1/2 (MM) | 13 1/2 (MM) | 14 10/19 (MM) | 15 7/11 (MM) |

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 9/29/18.