Problem of the Month (April 2020)

What is the smallest square that contains n non-overlapping objects, where the objects are either squares of side 1 or equilateral triangles of side 3/2?

ANSWERS

Maurizio Morandi sent improvements. Then in 2026, Yanis Gomes sent improvements and extended the table.

Here are the best known solutions:

n=1

s = 1 		n=2

s = 3√6/4 = 1.837+ 		n=3, 4

s = 2 		n=5

s = 2.568+
n=6

s = (19+27√3–3√15)/19 = 2.84981+ (YG) 		n=7, 8, 9

s = 3 		n=10

s = 3.469+ (MM) 		n=11

s = (43–2√3)/11 = 3.594+ (MM)
n=12

s = 3.78371+ (YG) 		n=13

s = (6+√3)/2 = 3.866+ 		n=14, 15, 16

s = 4 		n=17

s = 3+3√3/4 = 4.299+
n=18

s = 4.46279+ (YG) 		n=19

s = 2+3√3/2 = 4.598+ 		n=20

s = 4.644+ (MM) 		n=21

s = 4.86344+ (YG)
n=22

s = 4.97429+ (YG) 		n=23, 24, 25

s = 5 		n=26

s = 4+3√3/4 = 5.299+ 		n=27

s = 5.311+ (MM)
n=28

s = (125+11√3)/26 = 5.54048+ (YG) 		n=29

s = 3+3√3/2 = 5.598+ 		n=30

s = 5.692+ (MM) 		n=31

s = 5.78759+ (YG)
n=32

s = 5.89713+ (YG) 		n=33

s = 5.97777+ (YG) 		n=34, 35, 36

s = 6 		n=37

s = 6.29903+ (YG)
n=38

s = 5+3√3/4 = 6.299+ 		n=39

s = 6.40570+ (YG) 		n=40

s = 6.53309+ (YG) 		n=41

s = 4+3√3/2 = 6.598+
n=42

s = 6.64608+ (YG) 		n=43

s = (60–23√3)/3 = 6.72094+ (YG) 		n=44

s = 6.79477+ (YG) 		n=45

s = (12+9√3)/4 = 6.89711+ (YG)
n=46

s = 6.97980+ (YG) 		n=47, 48, 49

s = 7 		n=50

s = 7.29390+ (YG)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/12/20.