(October 2013)

Given such a digraph, can it be realized by a chess position in which the points represent chess pieces (no pawns), and there is an arrow from A to B only if piece A attacks piece B? If so, what is the smallest chessboard (in terms of area) on which this can occur?

The same questions can be asked for 3-regular digraphs, though they may be hard to count, and even harder to realize with chess positions. Can you show that no 4-regular digraph is realizable with a chess position?

(Richard Sabey) |

(Andrew Bayly) |

(Andrew Bayly) | (Joe DeVincentis) | (Andrew Bayly) | (Andrew Bayly) |

(Andrew Bayly) | (Andrew Bayly) | (Andrew Bayly) | (Maurizio Morandi) |

(Andrew Bayly) | (Andrew Bayly) | (Andrew Bayly) |

(Andrew Bayly) | (Andrew Bayly) | (George Sicherman) | (George Sicherman) |

(Andrew Bayly) | (Andrew Bayly) | (Andrew Bayly) | (Andrew Bayly) |

Joe DeVincentis showed that the number of 1-regular digraphs is A002865 at the OEIS.

Joe Devincentis showed that every 1-regular digraph can be realized as a chess position.

(Maurizio Morandi) | none |

(Maurizio Morandi) | (Andrew Bayly) | (Andrew Bayly) |

none (Joe DeVincentis) | (Maurizio Morandi) | none | (Maurizio Morandi) |

(Maurizio Morandi) | ? | (Maurizio Morandi) | ? |

? | ? | (Mark Thompson) | none (Mark Thompson) | (Joe DeVincentis) | (Mark Thompson) |

(Dave Langers) | (Dave Langers) | (Dave Langers) | (Dave Langers) | (Dave Langers) | (Dave Langers) |

Here are some 3-regular graphs realized by only knights, one of which comes from February 2007 Math Magic:

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) |

Most 3-regular digraphs with one or more directed edges can not be realized with chess positions. Here are a few that can be, using only queens and knights. Can you find any others?

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

Here are some others that use different pieces too:

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Andrew Bayly) | (Maurizio Morandi) |

(Geoff Exoo) | (Geoff Exoo) | (Joe DeVincentis) | (Maurizio Morandi) |

(Joe DeVincentis)

Here are some with directed edges, using queens and knights:

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Maurizio Morandi) | (Maurizio Morandi) |

(Maurizio Morandi) | (Andrew Bayly) |

Joe DeVincentis showed there are no 5-regular graphs or digraphs realizable with the usual chess pieces. What fairy chess pieces used in the August 2005 Math Magic have 5-regular realizations? George Sicherman found the positions below using amazons and archbishops:

(George Sicherman) | (George Sicherman) |

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/28/13.