We can illustrate this identity for a shape S by tiling a copy of S scaled by a factor of k with: n copies of S, (n-1) copies of S scaled by a factor of 2, ... up to 1 copy of S scaled by a factor of n. These are called **anti-partridge tilings**. Clearly the n=1 k=1 case is trivial. Anti-partridge tilings for n=6 of squares and equilateral triangles were found a decade ago. These can be stretched to yield anti-partridge tilings for n=6 of all parallelograms and triangles. What anti-partridge tilings can you find for n=6 or n=25?

Some rare shapes can have anti-partridge tilings for other values of n. There are apparently rectangle solutions for all values of n. Can you find some? There are also solutions for n=4 and n=10 using triangles. Can you find them? What other anti-partridge tilings can you find for non-integer values of k?

Patrick Hamlyn claimed to find all the solutions for the 30-60-90 triangle for n=6, but George Sicherman found many more.

Here are the known non-trivial anti-partridge tilings for n=6:

(Colin Singleton, 1996) | (Bob Wainwright, 2000) |

(Erich Friedman, 2002) | (Patrick Hamlyn, 2002) |

(Erich Friedman, 2002) | (Erich Friedman, 2002) | (Patrick Hamlyn, 2002) |

(Patrick Hamlyn) |

(Patrick Hamlyn) |

(George Sicherman) |

George Sicherman found many rectangles for other values of n.

Joe DeVincentis showed there are rectangle solutions for all n not equal to 1 mod 4 by stacking squares of size n+1-i above those of size i (like n=2 through 4 below).

Here are the known non-trivial anti-partridge tilings for other values of n:

n=2 | n=3 | n=4 | n=5 |

n=6 | n=7 | n=7 | n=7 |

n=7 | n=8 | n=8 | n=8 |

n=8 | n=8 | n=8 |

n=9 | n=9 | n=9 |

n=10 | n=10 |

n=4 |

n=10 |

Patrick Hamlyn likes tiling things on other surfaces too. He found this anti-partridge tiling for the bent triomino on a Möbius strip:

Patrick Hamlyn also found anti-partridge-like tilings to illustrate the following identities:

Claudio Baiocchi found this anti-partridge-like tiling:

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 7/2/07.