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? If we use the greedy algorithm, every golden rectangle or square used fills a factor of ϕ-1 of the area. Can we improve upon this?

ANSWERS

Maurizio Morandi and Sigvart Brendberg 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 = 3.876+ (SB) 		20

s = 3.963+ (MM) 		21-24

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

s = 4.834+ (MM)
32

s = 4.921+(MM) 		33

s = 8/ϕ 		34

s = 4.969+ (MM) 		35-40

s = 5

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/27/24.