(March 2013)

What are the answers for large m and n?

It is easy to show:

f(1,n) = n/2

f(2,n) = n/3

f(3,n) = n/4 for n≥2

f(4,n) = n/5 for n≥3

f(5,n) = n/5 for n≥12

f(6,1) = 3

f(6,6n) = n

f(6,6n+i) = n+2 for 1≤i≤4

f(6,6n+5) = n+3

Joe DeVincentis showed that f(7,6n+i) = 2n+2 for 1≤i≤2 and f(7,6n+i) = 2n+3 for 3≤i≤6.

Joe DeVincentis conjectured that f(8,3n-i) = n+2 for 0≤i&le2 and n≥10. He showed this pattern is eventually optimal, with limiting efficiency 24.

Joe DeVincentis also showed that the width 9 rectangles eventually have period 10 with limiting efficiency 30.

What is the eventual behavior of f(m,n) for larger m?

m\n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 9 | 10 | 10 | 11 | 11 | |||||||||||||

2 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 9 | 9 | 9 | 10 | 10 | 10 | 11 | 11 | 11 | ||

3 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 9 | 9 | 9 |

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

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

6 | 3 | 2 | 2 | 2 | 3 | 1 | 3 | 3 | 3 | 3 | 4 | 2 | 4 | 4 | 4 | 4 | 5 | 3 | 5 | 5 | 5 | 5 | 6 | 4 | 6 | 6 | 6 | 6 | 7 | 5 | 7 | 7 | 7 | 7 | 8 |

7 | 4 | 3 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | GS | JD | JP | JG | 7 | 7 | JD | JD | JD | JD | 9 | 9 | JD | JD | JD | JD | 11 | 11 | JD | JD | |||||

8 | 4 | 3 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | JD | 6 | 6 | 6 | 7 | 8 | 8 | 8 | 8 | 9 | 10 | 10 | 10 | 10 | 11 | JD | ||||||||

9 | 5 | 3 | 3 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 7 | JP | 8 | 8 | 8 | 9 | 9 | JD | 10 | 10 | JD | 11 | JD | JD | JD | |||||

10 | 5 | 4 | 3 | 2 | 2 | 3 | 5 | 4 | 4 | 4 | JP | 5 | 6 | 6 | 6 | JP | 7 | 7 | 8 | 8 | 8 | 8 | 9 | 9 | JD | 10 | 10 | 11 | 11 | 11 | |||||

11 | 6 | 4 | 3 | 3 | 4 | 4 | GS | 5 | 5 | JP | JD | 5 | JD | JD | JD | JD | JD | 7 | JD | 8 | 8 | 8 | 9 | JP | JD | JP | JP | JD | 11 | JD | |||||

12 | 6 | 4 | 3 | 3 | 3 | 2 | JD | JD | 5 | 5 | 5 | 4 | 7 | JD | 7 | 7 | 7 | 6 | JD | 9 | 9 | 9 | 9 | 8 | 9 | 11 | 11 | 11 | 11 | 10 | 11 | ||||

13 | 7 | 5 | 4 | 3 | 3 | 4 | JP | 6 | 6 | 6 | JD | 7 | JG | JD | JD | JD | JD | JD | JD | JD | JD | JD | JD | JD | |||||||||||

14 | 7 | 5 | 4 | 3 | 3 | 4 | JG | 6 | 6 | 6 | JD | JD | JD | JD | JD | JD | JD | JG | JD | JD | 11 | JD | JD | JD | |||||||||||

15 | 8 | 5 | 4 | 3 | 3 | 4 | 7 | 6 | 6 | 6 | JD | 7 | JD | JD | 9 | EF | 10 | JG | JD | JD | JD | JD | JD | JD | |||||||||||

16 | 8 | 6 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | JP | JD | 7 | JD | JD | EF | JG | JD | JP | JD | JD | |||||||||||||||

17 | 9 | 6 | 5 | 4 | 4 | 5 | JD | 8 | JP | 7 | JD | 7 | JD | JD | 10 | JD | JG | JP | JD | ||||||||||||||||

18 | 9 | 6 | 5 | 4 | 4 | 3 | JD | 8 | 8 | 7 | 7 | 6 | JD | JG | JG | JP | JP | 9 | JD | ||||||||||||||||

19 | 10 | 7 | 5 | 4 | 4 | 5 | JD | 8 | 8 | 8 | JD | JD | JD | JD | JD | JD | JD | JD | |||||||||||||||||

20 | 10 | 7 | 5 | 4 | 4 | 5 | JD | 8 | 8 | 8 | 8 | 9 | JD | JD | JD | JD | |||||||||||||||||||

21 | 11 | 7 | 6 | 5 | 5 | 5 | 9 | 9 | 9 | 8 | 8 | 9 | JD | 11 | JD | ||||||||||||||||||||

22 | 11 | 8 | 6 | 5 | 5 | 5 | 9 | 10 | 9 | 8 | 8 | 9 | JD | JD | |||||||||||||||||||||

23 | 8 | 6 | 5 | 5 | 6 | JD | 10 | JD | 9 | 9 | 9 | JD | JD | JD | |||||||||||||||||||||

24 | 8 | 6 | 5 | 5 | 4 | JD | 10 | 10 | 9 | JP | 8 | JD | JD | JD | |||||||||||||||||||||

25 | 9 | 7 | 5 | 5 | 6 | JD | 10 | 10 | JD | JD | 9 | JD | |||||||||||||||||||||||

26 | 9 | 7 | 6 | 6 | 6 | JD | 11 | JD | 10 | JP | 11 | ||||||||||||||||||||||||

27 | 9 | 7 | 6 | 6 | 6 | 11 | JD | 11 | 10 | JP | 11 | ||||||||||||||||||||||||

28 | 10 | 7 | 6 | 6 | 6 | 11 | JD | 11 | JD | 11 | |||||||||||||||||||||||||

29 | 10 | 8 | 6 | 6 | 7 | JD | JD | 11 | 11 | 11 | |||||||||||||||||||||||||

30 | 10 | 8 | 6 | 6 | 5 | JD | JD | 11 | JD | 10 | |||||||||||||||||||||||||

31 | 11 | 8 | 7 | 7 | 7 | 11 | |||||||||||||||||||||||||||||

32 | 11 | 8 | 7 | 7 | 7 | ||||||||||||||||||||||||||||||

33 | 11 | 9 | 7 | 7 | 7 | ||||||||||||||||||||||||||||||

34 | 9 | 7 | 7 | 7 | |||||||||||||||||||||||||||||||

35 | 9 | 7 | 7 | 8 |

JP = Jon Palin

GS = George Sicherman

JG = Jeremy Galvagni

EF = Erich Friedman

Jeremy Galvagni tried to cover a 100×100 square to see whether small, medium, or large circles were most efficient. By using a large circle, he managed a covering with 145 circles. Joe DeVincentis improved this to 129 circles, shown below. Can anyone do better?

Jeremy Galvagni also defined the covering efficiency of a covering as the area covered per circle. He notes that the multiples of the 6×6 packing of one circle all have efficiency 36. He asked for the smallest covering that is more efficient than this. Joe DeVincentis found a covering of a 22×22 square with a higher efficiency, shown below.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/31/13.