# Pairwise Touching Polyominoes In Boxes

#### We say a collection of polyominoes is pairwise touching if every polyomino touches every other along at least one face. If we cut up an n x m x p box into polyominoes, what is the maximum number of pairwise touching polyominoes we can get? The tables below show the best-known bounds. Click on the links for pictures.

Pairwise Touching Polyominoes in n x m x 1 Box
n \ m |
1 |
2 |
3 |
4 |
5 |

1 |
1 |
2 |
2 |
2 |
2 |

2 |
2 |
3 |
3 |
3 |
3 |

3 |
2 |
3 |
4 |
4 |
4 |

4 |
2 |
3 |
4 |
4 |
4 |

5 |
2 |
3 |
4 |
4 |
4 |

Pairwise Touching Polyominoes in n x m x 2 Box
n \ m |
1 |
2 |
3 |
4 |
5 |

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

2 |
3 |
4 |
4 |
4 |
4 |

3 |
3 |
4 |
5 |
6 |
6 |

4 |
3 |
4 |
6 |
7 |
7,8 |

5 |
3 |
4 |
6 |
7,8 |
7,10 |

Pairwise Touching Polyominoes in n x m x 3 Box
n \ m |
1 |
2 |
3 |
4 |
5 |

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

2 |
3 |
4 |
5 |
6 |
6 |

3 |
4 |
5 |
5 |
6 |
7 |

4 |
4 |
6 |
6 |
8 |
8,10 |

5 |
4 |
6 |
7 |
8,10 |
9,13 |

Pairwise Touching Polyominoes in n x m x 4 Box
n \ m |
1 |
2 |
3 |
4 |
5 |

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

2 |
3 |
4 |
6 |
7 |
7,8 |

3 |
4 |
6 |
6 |
8 |
8,12 |

4 |
4 |
7 |
8 |
9,12 |
9,16 |

5 |
4 |
7,8 |
8,12 |
9,16 |
10,18 |