Can you find solutions with few pieces for n=3? What about larger square boards? What about rectangular boards? Are the answers different if you are allowed to use both black and white pieces?

Joe DeVincentis showed that:

These bounds seem to be sharp for k=2, k=7, and k=8.

Here are the best known solutions.

1 | 2 | 3 |

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

1 | 2 | 3 (Joe DeVincentis) | 4 |

5 (Heinrich Hemme) | 6 | 7 | 8 |

9 (James Wilson) | 10 | 11 | 12 |

1 | 2 | 3 (Dirk Riehm) | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 (James Wilson) | 12 |

13 (James Wilson) | 14 | 15 | 16 |

1 | 2 | 3 (Joe DeVincentis) | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 (James Wilson) | 12 |

13 (James Wilson) | 14 | 15 | 16 |

1 (Joe DeVincentis) | 2 | 3 (Joe DeVincentis) | 4 |

5 | 6 | 7 | 8 |

9 (James Wilson) | 10 (James Wilson) | 11 | 12 (James Wilson) |

13 (James Wilson) | 14 | 15 | 16 |

Here are the best known results when every square must be attacked an equal number of times.

1 | 2 (James Wilson) | 3 |

1 | 2 (Joe DeVincentis) | 3 |

1 | 2 (Bernd Rennhak) | 3 (Bernd Rennhak) | 4 |

1 (Bernd Rennhak) | 2 | 3 (Bernd Rennhak) | 4 (Bernd Rennhak) |

1 | 2 (Bernd Rennhak) | 3 (Bernd Rennhak) | 3 (Bernd Rennhak) |

1 | 2 | 3 (Bernd Rennhak) |

1 (Bernd Rennhak) |

1 (Bernd Rennhak) |

Bernd Rennhak considered the problem of attacking every square on an 4×n chessboard exactly k times.

1 | 2 | 3 | 4 |
---|---|---|---|

(Bernd Rennhak) | (Bernd Rennhak) | (Bernd Rennhak) | (Bernd Rennhak) |

(Bernd Rennhak) | (Bernd Rennhak) | (Bernd Rennhak) | (Bernd Rennhak) |

(Bernd Rennhak) | (Bernd Rennhak) | ||

(Bernd Rennhak) | |||

(Bernd Rennhak) |

Bernd Rennhak also considered the problem of attacking every square on triangular chessboards exactly k times.

1 | 2 | 3 |
---|---|---|

(Bernd Rennhak) | (Bernd Rennhak) | |

(Bernd Rennhak) | (Bernd Rennhak) | (Bernd Rennhak) |

(Bernd Rennhak) | (Bernd Rennhak) | (Bernd Rennhak) |

(Bernd Rennhak) | (Bernd Rennhak) |

Bernd Rennhak has these pages with more information.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/13/07.