We can also look for **Border Sub-Partridge Tilings**, a tiling of a rectangle with squares with a border of width 1 around each square. For example, a {1,3,4,5,7} border tiling is shown below. What border sub-partridge tilings exist?

The second is {2,4,6,...2n}, as shown by the following tiling:

The third is {4,8,24,60,...4F_{n}F_{n+1}} (where F_{n} is the n^{th} Fibonacci number), as shown by the following tiling:

Jeremy Galvagni showed that {a^{2}bc,ab^{2}c} tilings exist. He conjectured that these are the only sub-partridge tilings using only two sizes of squares.

Luke Pebody showed that this conjecture isn't true by finding the {35,70} tiling below. He and Emilio Schiavi showed that an {n,2n} tiling does not exist precisely when n is an odd prime or n=1, 9, 15, 21, 25, 27, or 33.

Patrick Hamlyn found many sub-partridge tilings with his tiling program.

Patrick Hamlyn and Jeremy Galvagni noted that border tilings are equivalent to tiling n-1 copies of each square of side n without borders.

Claudio Baiocchi notes that if there is a {a_{1}, a_{2}, . . . a_{n}} tiling, then there is a {k a_{1}, k a_{2}, . . . k a_{n}} tiling for every k, and that for every {a_{1}, a_{2}, . . . a_{n}}, there exists some k so that there is a {k a_{1}, k a_{2}, . . . k a_{n}} tiling.

Largest Square | Number of Sizes | Tilings |
---|---|---|

5 | 2 | {2,5} |

7 | 5 | {1,3,4,5,7} |

6 | {2,3,4,5,6,7} | |

8 | 7 | {2,3,4,5,6,7,8} |

8 | {1,2,3,4,5,6,7,8} | |

9 | 2 | {4,9}, {5,9} |

3 | {5,8,9} | |

6 | {3,4,5,6,8,9} | |

7 | {1,2,4,6,7,8,9} | |

8 | {1,2,3,4,5,7,8,9}, {1,2,4,5,6,7,8,9} | |

9 | {1,2,3,4,5,6,7,8,9} | |

10 | 8 | {1,2,3,5,6,7,9,10} |

11 | 2 | {3,11} |

4 | {2,5,8,11} | |

7 | {1,3,4,5,7,9,11} | |

8 | {1,2,3,4,5,6,9,11} | |

13 | 2 | {6,13} |

14 | 2 | {5,14}, {9,14} |

3 | {5,9,14} | |

4 | {2,8,11,14} | |

15 | 4 | {5,8,9,15} |

16 | 3 | {3,11,16} |

17 | 2 | {8,17} |

4 | {5,8,9,17}, {8,9,14,17} | |

18 | 3 | {8,17,18} |

19 | 2 | {4,19} |

3 | {4,9,19} |

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 1/31/06.