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?


ANSWERS

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.