Problem of the Month (December 2019)

Given a positive integer n, how can be pack n squares inside a square to maximize the product of their side lengths? What about circles or equilateral triangles in squares?

ANSWERS

Answers were received from Maurizio Morandi and Marc Lapierre.

Squares in Squares
n=1

P = 1 		n=2

P = 1/22 = .25 		n=3

P = 1/23 = .125 		n=4

P = 1/24 = 6.25 × 10–2 		n=5

P = 1/3322 = 9.25 × 10–3
n=6

P = 2/36 = 2.74 × 10–3 		n=7

P = 1/211 = 4.88 × 10–4 		n=8

P = 1/38 = 1.52 × 10–4 		n=9

P = 1/39 = 5.08 × 10–5 		n=10

P = 1/2836 = 5.35 × 10–6
n=11

P = 34/226 = 1.20 × 10–6 		n=12

P = 9/226 = 1.34 × 10–7
(MM) 		n=13

P = 1/225 = 2.98 × 10–8 		n=14

P = 29/323 = 5.43 × 10–9 		n=15

P = 1/230 = 9.31 × 10–10
n=16

P = 1/232 = 2.32 × 10–10 		n=17

P = 1/55224 = 1.90 × 10–11 		n=18

P = 218/51839 = 3.49 × 10–12 		n=19

P = 212/51936 = 2.94 × 10–13
(MM) 		n=20

P = 34/52024 = 5.30 × 10–14
n=21

P = 1/39233 = 5.91 × 10–15 		n=22

P = 2/522 = 8.38 × 10–16 		n=23

P = 316/278 = 1.42 × 10–16 		n=24

P = 1/524 = 1.67 × 10–17 		n=25

P = 1/525 = 3.35 × 10–18
n=26

P = 1/66520 = 2.24 × 10–19 		n=27

P = 516/327259 = 3.47 × 10–20 		n=28

P = 512/328252 = 2.36 × 10–21
(MM) 		n=29

P = 1/338211 = 3.61 × 10–22 		n=30

P = 29/53039 = 2.79 × 10–23

Circles in Squares
n=1

P = 1/2 = .5 		n=2

P = 3/2 – √2 = 8.57 × 10–2 		n=3

P = 1.80 × 10–2
(MM) 		n=4

P = 1/28 = 3.90 × 10–3 		n=5

P = 4.28 × 10–4
(ML)
n=6

P = 5.20 × 10–5
(ML) 		n=7

P = 6.58 × 10–6
(MM) 		n=8

P = 9.02 × 10–7 		n=9

P = 1/3929 = 9.92 × 10–8 		n=10

P = 8.15 × 10–9
(MM)
n=11

P = 8.03 × 10–10
(ML) 		n=12

P = 8.38 × 10–11 		n=13

P = 6.13 × 10–12 		n=14

P = 5.80 × 10–13 		n=15

P = 5.00 × 10–14
(ML)
n=16

P = 1/816 = 4.97 × 10–14 		n=17

P = 2.56 × 10–16
(ML) 		n=18

P = 1.94 × 10–17
(ML) 		n=19

P = 9.78 × 10–19
(ML) 		n=20

P = 9.38 × 10–20
n=21

P = 7.42 × 10–21 		n=22

P = 3.37 × 10–22 		n=23

P = 4.26 × 10–23 		n=24

P = 2.00 × 10–24 		n=25

P = 1/1025 = 1.00 × 10–25

Triangles in Squares
n=1

P = (√3–1)/2 = 1.03 		n=2

P = 2/3 = .666 		n=3

P = 6√2–10√(2/3) = .320 		n=4

P = 63/4–9√3 = .161 		n=5

P = 5.42 × 10–2
n=6

P = 2.11 × 10–2 		n=7

P = 1/27 = 7.81 × 10–3
(MM) 		n=8

P = 2.79 × 10–3 		n=12

P = 1.28 × 10–5

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/1/19.