What are the values of m_n(r) for non-square values of n? A much harder question is to find M_n(r), the smallest value of m so that so that an m x mr rectangle contains n non-overlapping 1 x r rectangles that might be tilted. For example, when r > 1.9762, M₂(r) < m₂(r):

What are the values of M_n(r) for small values of n?

Here are the piecewise definitions for m_n(r) for small n:

If	r ≥ 1	r ≥ √2	r ≥ 2
n=2
then	m2(r) = 2/r	m2(r) = r	m2(r) = 2

If	r ≥ 1	r ≥ 3/2	r ≥ √3	r ≥ 2
n=3
then	m3(r) = 2	m3(r) = 3/r	m3(r) = r	m3(r) = 2

If	r ≥ 1	r ≥ (√13-1)/2	r ≥ (√5+1)/2	r ≥ 2	r ≥ √5	r ≥ 3
n=5
then	m5(r) = 3/r	m5(r) = r + 1	m5(r) = 2 + 1/r	m5(r) = 5/r	m5(r) = r	m5(r) = 3

If	r ≥ 1	r ≥ √(3/2)	r ≥ (√3+1)/2	r ≥ 3/2	r ≥ (√17-1)/2	r ≥ 2	r ≥ √6	r ≥ 3
n=6
then	m6(r) = 3/r	m6(r) = 2r	m6(r) = 2 + 1/r	m6(r) = 4/r	m6(r) = r + 1	m6(r) = 6/r	m6(r) = r	m6(r) = 3

If	r ≥ 1	r ≥ 4/3	r ≥ √2	r ≥ 3/2	r ≥ √3	r ≥ 2	r ≥ 7/3	r ≥ √7	r ≥ 3
n=7
then	m7(r) = 3	m7(r) = 4/r	m7(r) = 2r	m7(r) = 1 + 3/r	m7(r) = r + 1	m7(r) = 3	m7(r) = 7/r	m7(r) = r	m7(r) = 3

If	r ≥ 1	r ≥ 4/3	r ≥ √2	r ≥ 3/2	r ≥ 8/3	r ≥ √8	r ≥ 3
n=8
then	m8(r) = 3	m8(r) = 4/r	m8(r) = 2r	m8(r) = 3	m8(r) = 8/r	m8(r) = r	m8(r) = 3

If	r ≥ 1	r ≥ √5-1	r ≥ √2	r ≥ 3/2	r ≥ √(5/2)	r ≥ (√17+3)/4
n=10
	m10(r) = 4/r	m10(r) = 2 + r	m10(r) = 2 + 2/r	m10(r) = 5/r	m10(r) = 2r	m10(r) = 3 + 1/r
	r ≥ 2	r ≥ (√13+1)/2	r ≥ 1+√2	r ≥ 3	r ≥ √10	r ≥ 4
then	m10(r) = 2 + 3/r	m10(r) = 1 + r	m10(r) = 3 + 1/r	m10(r) = 10/r	m10(r) = r	m10(r) = 4

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 6/7/04.