Is there a polyomino floe with a square that contains the number 5? Are there polyomino floes containing arbitrarily large integers? What is the smallest size of an integer floe that contains a square labeled n?

What about this problem in 1 dimension or 3 dimensions? What about other lattices, like the triangular or hexagonal lattice? What about other face-transitive tessellations involving regular polygons?

Polyominoes

n=1 | n=2, 4 | n=3 | n=5 (Kang Jin Cho) | n=6 |

n=7, 8 | n=9 | n=10, 12, 14, 16 (George Sicherman) |

Set | Samples | Areas | |
---|---|---|---|

{1} | 4, even n ≥ 8 | ||

{1, 2, 3} | (Bryce Herdt) | (Jon Palin) | 20, 40+10n, 52+6n, 84+6n, 116+6n, so all even n ≥ 106 |

{1, 2, 3, 4} | (George Sicherman) | 13, 22, 31, even n ≥ 32, odd n ≥ 41 | |

{2, 3, 4} | 76+18m+40n+46p+56q, so all even n ≥ 202 | ||

{1, 2, 3, 4, 5, 6} | (Kang Jin Cho) | 68, 204 + 68n | |

{1, 2, 3, 4, 6} | (Kang Jin Cho) | 244 + 6m + 190n, 260 + 6m + 190n, 418 + 6m so all even n ≥ 250 (Bryce Herdt) | |

{1, 2, 3, 4, 5, 6, 7} | (George Sicherman) | 156n (Bryce Herdt) |

Polyiamonds

n=1, 2 | n=3 | n=4 (George Sicherman) | n=5 | n=6 | n=7 | n=8, 11 |

n=9, 12 (George Sicherman) | n=10 (George Sicherman) | n=13 (George Sicherman) | n=14, 17, 20 (George Sicherman) | n=15, 18 (George Sicherman) |

n=16, 25, 26 (George Sicherman) | n=19, 21, 24, 27, 29, 31, 34 (George Sicherman) | n=22 ? | n=23 ? | n=30 (George Sicherman) |

Set | Samples | Areas | |
---|---|---|---|

{2} | 6, even n ≥ 12 | ||

{1, 2, 3, 4, 5} | (George Sicherman) | 33 + 12n |

Some other small polyiamond floes are shown below. Are there some with arbitrarily many holes?

(George Sicherman) | (George Sicherman) | (George Sicherman) |

(George Sicherman) | (George Sicherman) | (George Sicherman) |

(George Sicherman) | (George Sicherman) | (George Sicherman) | (George Sicherman) |

Polyhexes

n=1 | n=2, 3 | n=4 ? |

Set | Samples | Areas | |
---|---|---|---|

{1} | (Bryce Herdt) | 24, 32, 34, 36, even n ≥ 40 | |

{1, 2, 3} | (George Sicherman) | multiples of 10 |

These last pictures show that there are polyhex floes with arbitrarily large holes and arbitrarily many holes.

Triangular/Hexagonal Grid

n=1, 2 | n=3 | n=4, 5 | n=6 |

Set | Samples | Areas | |
---|---|---|---|

{1, 2} | 7 + 3n, 15 + 3n, 20 + 3n, so all n ≥ 18 | ||

{2} | 18, 24 + 3n (Bryce Herdt) | ||

{1, 2, 3} | 10, 30, ? |

Square/Octagonal Grid

n=1 | n=2 ? |

Set | Samples | Areas | |
---|---|---|---|

{1} | 20 + 8m + 10n, 34 + 8m + 10n, so all even n ≥ 34 |

The smallest floe known on the triangular/dodecagonal lattice has 96 cells, shown below.

(Bryce Herdt)

Bryce Herdt considered and completely classified the 1-dimensional floes. All lines of cells are floes, such as 11, 232, 3553, 47874, and their values are quadratic in position.

Bryce Herdt also showed that if we omit one face of a platonic solid in 3-dimensions, the remaining faces form a 2-dimensional floe. A tetrahedron has faces with 2's, a cube has 4's and a 5, an octahedron has 6's, 8's, and a 9, a dodecahedron has 10's, 13's, and a 14, and an icosahedron has 18's, 26's, 31's, 33's, and a 34. He also gave many other subsets of platonic solid faces that served as floes.

Jon Palin considered cubes in 3-dimensions, with 6 possible directions of movement. He showed that any loop of 1's in 2-dimensions has a height 2 analog of cubes. The same argument shows there are arbitrarily large polycube floes in any dimension.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/1/16.