Problem of the Month(February 2019)

What is the smallest square that can contain n non-overlapping unit golden rectangles (1×ϕ, where ϕ=(1+√5)/2)? What is the smallest golden rectangle that can contain these squares? What is the smallest golden rectangle that can contain n unit squares?

How full can we pack a unit square with golden rectangles of any size? Similarly, how full can we pack a unit golden rectangle with squares of any size?

Maurizio Morandi sent solutions.

Best-Known Packings of Golden Rectangles in Squares
 1 s = ϕ = 1.618+ 2 s = 2 3-4 s = 1 + ϕ = 2.618+ 5-6 s = 2ϕ = 3.236+ 7-8 s = 2 + ϕ = 3.618+ 9-10 s = 1 + 2ϕ = 4.236+ 11-12 s = 3 + ϕ = 4.618+ 13 s = 3ϕ = 4.854+ 14-15 s = 5 16 s = 2 + 2ϕ = 5.236+ 17-18 s = 4 + ϕ = 5.618+ 19-20 s = 1 + 3ϕ = 5.854+ 21 s = 6 22-24 s = 3 + 2ϕ = 6.236+ 25-26 s = 5 + ϕ = 6.618+ 27-28 s = 2 + 3ϕ = 6.854+ 29 s = 7 30-32 s = 4 + 2ϕ = 7.236+ 33-35 s = 6 + ϕ = 7.618+ 36-37 s = 3 + 3ϕ = 7.854+ (MM) 38 s = 8 (MM) 39-40 s = 5ϕ = 8.090+

Best-Known Packings of Golden Rectangles in Golden Rectangles
 1 s = 1 2 s = ϕ = 1.618+ 3 s = 3/ϕ = 1.854+ 4 s = 2 5-6 s = 1 + ϕ = 2.618+ 7 s = 1 + 3/ϕ = 2.854+ 8-9 s = 3 10 s = 2ϕ = 3.236+ 11 s = 1 + 4/ϕ = 3.472+ 12 s = 2 + ϕ = 3.618+ 13 s = 6/ϕ = 3.708+ 14-16 s = 4 17 s = 1 + 2ϕ = 4.236+ 18 s = 7/ϕ = 4.326+ 19-20 s = 3 + ϕ = 4.618+ 21 s = 1 + 6/ϕ = 4.708+ 22 s = 3ϕ = 4.854+ 23-24 s = 8/ϕ = 4.944+ 25 s = 5 26 s = 2 + 2ϕ = 5.236+ 27 s = 1 + 7/ϕ = 5.326+ 28 s = 3 + 4ϕ = 5.472+ 29-31 s = 4 + ϕ = 5.618+ 32-33 s = 1 + 3ϕ = 5.854+ 34 s = 1 + 8/ϕ = 5.944+
 35-36 s = 6

Best-Known Packings of Squares in Golden Rectangles
 1 s = 1 2 s = 2/ϕ = 1.236+ 3 s = 3/ϕ = 1.854+ 4-6 s = 2 7 s = 2.450+ (MM) 8 s = 4/ϕ = 2.472+ 9 s = 2.830+ (MM) 10 s = 2.945+ (MM) 11-12 s = 3 13-15 s = 5/ϕ = 3.090+ 16 s = 3.589+ (MM) 17-18 s = 6/ϕ = 3.708+ 19 s = 3.889+ (MM) 20-24 s = 4 25 s = 4.294+ (MM) 26-28 s = 7/ϕ = 4.326+ 29 s = 4.676+ (MM) 30 s = 4.741+ (MM) 31-32 s = 8/ϕ = 4.944+ 33-40 s = 5

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/1/12.