(June 2016)

Let f(m,n) be the smallest k that works for a given m and n.

It is clear that f(1,n) = f(2,n) = n/3 and f(a+b,n) ≤ f(a,n) + f(b,n).

Joe DeVincentis showed the following general bounds for large n:

f(3,7n+3)≤2n+1

f(4,4n)≤n

f(5,9n–7)≤2n

f(5,9n–2)≤2n+1

f(6,5n–7)≤n

f(6,11n–3)≤2n+1

f(6,11n–9)≤2n

Triangles Covering m×n Rectangle

m\n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 |

2 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 |

3 | 1 | 1 | EF | 2 | 2 | 2 | 3 | 3 | 3 | EF | 4 | 4 | 4 | 5 | 5 | 5 | JD | 6 | 6 | 6 |

4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | JD | 3 | 3 | 3 | EF | 4 | 4 | 4 | EF | 5 | 5 | 5 | JD |

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

6 | 2 | 2 | 2 | 2 | 3 | 3 | EF | 4 | 4 | 4 | 4 | 4 | JD | 5 | 5 | 5 | 5 | JD | 6 | 6 |

7 | 3 | 3 | 3 | 2 | 3 | EF | 4 | 4 | 4 | JD | JD | JD | 4 | JD | JD | JD | JD | JD | JD | JD |

8 | 3 | 3 | 3 | JD | 4 | 4 | 4 | JD | JD | 5 | JD | JD | JD | JD | JD | JD | JD | JD | JD | 7 |

9 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | JD | 5 | 5 | JD | JD | JD | JD | JD | JD | JD | JD | JD | 7 |

10 | 4 | 4 | EF | 3 | 4 | 4 | JD | 5 | 5 | 5 | 5 | JD | JD | JD | JD | JD | JD | JD | JD | 7 |

11 | 4 | 4 | 4 | 3 | 4 | 4 | JD | JD | JD | 5 | JD | JD | JD | JD | JD | JD | JD | JD | JD | JD |

12 | 4 | 4 | 4 | EF | 4 | 4 | JD | JD | JD | JD | JD | JD | JD | JD | JD | JD | JD | JD | JD | JD |

13 | 5 | 5 | 4 | 4 | 5 | JD | 4 | JD | JD | JD | JD | JD | 8 | JD | JD | JD | JD | JD | JD | JD |

14 | 5 | 5 | 5 | 4 | 5 | 5 | JD | JD | JD | JD | JD | JD | JD | |||||||

15 | 5 | 5 | 5 | 4 | 5 | 5 | JD | JD | JD | JD | JD | JD | JD | |||||||

16 | 6 | 6 | 5 | EF | 6 | 5 | JD | JD | JD | JD | JD | JD | JD | |||||||

17 | 6 | 6 | JD | 5 | 6 | 5 | JD | JD | JD | JD | JD | JD | JD | |||||||

18 | 6 | 6 | 6 | 5 | 6 | JD | JD | JD | JD | JD | JD | JD | JD | |||||||

19 | 7 | 7 | 6 | 5 | 6 | 6 | JD | JD | JD | JD | JD | JD | JD | |||||||

20 | 7 | 7 | 6 | JD | JD | 6 | JD | 7 | 7 | 7 | JD | JD | JD |

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