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≤2 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?

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.