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

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

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