n=16 (Karl Scherer) | n=18 (Patrick Hamlyn) | n=22 (Patrick Hamlyn) | n=23 (Karl Scherer) | n=24 (Patrick Hamlyn) |

n=25 (Karl Scherer) | n=26 (Karl Scherer) | n=27 (Karl Scherer) | n=28 (Karl Scherer) |

This month we ask which other reptiles have no-touch tilings, and for those that do, what are their no-touch numbers?

Can you show that equilateral triangles and cubes do not have no-touch tilings? Can you find a no-touch tiling of a bent triomino? Can you show that that dominoes have no-touch tilings for n≥5? What are the no-touch numbers of other rectangles? Other polyominoes? Polyiamond or polytan trapezoids? Other shapes?

We continue the recent tradition of offering a $10 prize to solver offering the best solutions.

Tino Jonker noted that all integer-sided right triangles have non-touch tilings, because the altitude to the hypotenuse divides the triangle into two similar triangles. He was also able to prove that almost all tritans have no-touch tilings.

Shape | Known No-Touch Numbers | Unknown |
---|---|---|

16, 18, 22+ | ||

5+ | ||

14+ | ||

34, 44 | 28-33, 35-43, 45+ | |

16, 18, 21+ | ||

16, 18-20, 22+ | ||

15, 16, 18+ | ||

7, 10-46 | 47+ | |

23+ | ||

none | ||

7, 10, 11, 13+ | ||

19, 24, 25, 27-30 | 31+ | |

17, 18, 20 | 21+ | |

7, 10, 13-24 | 25+ | |

19 | 20+ |

Here are 1×2 rectangle no-touch tilings that do not come from square or smaller tilings:

n=5 | n=6 | n=7 | n=8 |

n=9 | n=11 | n=13 |

n=17 | n=19 |

Here are 1×3 rectangle no-touch tilings that do not come from square tilings:

n=14 (Patrick Hamlyn) | n=15 |

n=17 (Patrick Hamlyn) | n=19 (Patrick Hamlyn) |

n=20 | n=21 |

Here are 1×4 and 1×5 rectangle no-touch tilings that do not come from square tilings:

n=21 (Patrick Hamlyn) |

n=19 (Patrick Hamlyn) | n=20 (Patrick Hamlyn) |

Here are 2×3 rectangle no-touch tilings that do not come from square tilings:

n=15 (Patrick Hamlyn) | n=19 (Patrick Hamlyn) |

n=20 (Patrick Hamlyn) | n=21 (Patrick Hamlyn) |

Here are small bent triomino no-touch tilings:

n=34 (Patrick Hamlyn) | n=44 |

Here are small tritan no-touch tilings that do not come from smaller tilings:

n=7 (Patrick Hamlyn) | n=10 (Patrick Hamlyn) | n=11 (Patrick Hamlyn) | n=12 (Patrick Hamlyn) |

n=13 (Patrick Hamlyn) | n=15 (Patrick Hamlyn) | n=16 (Joseph DeVincentis) |

n=17 (Joseph DeVincentis) | n=18 (Patrick Hamlyn) |

n=19 (Joseph DeVincentis) | n=23 (Patrick Hamlyn) |

n=25 (Patrick Hamlyn) | n=27 (Patrick Hamlyn) |

n=29 (Joseph DeVincentis) | n=31 (Patrick Hamlyn) |

n=37 (Patrick Hamlyn) |

n=41 (Patrick Hamlyn) |

n=43 (Patrick Hamlyn) |

Here are small triamond no-touch tilings that do not come from smaller tilings:

n=7 (Patrick Hamlyn) | n=10 | n=11 (Patrick Hamlyn) | n=13 |

n=15 | n=16 |

n=17 | n=18 |

n=19 | n=22 |

Here are small straight pentiamond no-touch tilings:

n=19 (Patrick Hamlyn) | n=24 (Patrick Hamlyn) |

n=25 (Patrick Hamlyn) |

n=27 (Patrick Hamlyn) |

n=28 (Patrick Hamlyn) |

n=29 (Patrick Hamlyn) |

n=30 (Patrick Hamlyn) |

Here are small octiamond no-touch tilings:

n=17 (Patrick Hamlyn) | n=18 (Patrick Hamlyn) |

n=20 (Patrick Hamlyn) |

Here are small tridrafter no-touch tilings:

n=7 (Patrick Hamlyn) | n=10 (Patrick Hamlyn) | n=13 (Patrick Hamlyn) | n=15 (Patrick Hamlyn) | n=16 (Patrick Hamlyn) |

n=17 (Patrick Hamlyn) | n=18 (Patrick Hamlyn) | n=19 (Patrick Hamlyn) |

n=22 (Patrick Hamlyn) | n=23 (Patrick Hamlyn) | n=24 (Patrick Hamlyn) |

Here are small pentadrafter no-touch tilings:

n=19 (Patrick Hamlyn) |

And the winner of the $10 prize is Patrick Hamlyn.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 7/30/05.