Problem of the Month
(October 2012)

There are lots of packing problems in the Math Magic Packing Archive. But this month we feature the first 3-dimensional packing problem. What is the smallest cubical box that will fit n cylinders with radius and height 1 (tuna cans) ? What is the smallest cubical box that will fit n cylinders with diameter and height 1 (soup cans) ?

I suspect this is a pretty hard problem if we are allowed to tilt the cylinders, so I also offer the easier problem: What is the smallest cubical box that will fit n cylinders, each parallel to one of the sides of the box?

ANSWERS

Improvements were received by Jeremy Galvagni, David W. Cantrell, and Maurizio Morandi.

Here are the best-known packings of tuna cans. For each, a 3-dimensional view and a top view are provided.

2

s = 2
3

s = 2 + 8/√65 = 2.992+ (DC)
6

s = 3
7

s = 18/5 = 3.6 (DC) (MM)

8

s = 3.683+ (MM)

9

s = 3.890+ (DC)

10

s = 3.985+

16

s = 4

18

s = 2(7+√19)/5 = 4.543+ (MM)

19

s = 4.709+ (DC)

20

s = 4.816+ (DC)

21

s = 2 + 2√2 = 4.828+

22

s = 4.947+ (DC)

24

s = 4.959+ (DC)

30

s = 5

31

s = (5+√31)/2 = 5.283+

32

s = 2 + 12/√13 = 5.328+

33

s = 5.512+

34

s = 5.546+

35

s = 5.669+ (DC)

36

s = 5.714+ (DC)

37

s = 4 + √3 = 5.732+

40

s = 2 + √2 + √6 = 5.863+

41

s = 4(14+3√3)/13 = 5.906+

42

s = 5.970+ (DC)

44

s = 5.988+ (DC)

46

s = 5.999+ (DC)

54

s = 6

Here are the best-known packings of soup cans.

1

s = 2
2

s = 2 + √2 = 3.414+
3

s = 3 + 1/√2 = 3.707+
8

s = 4

10

s = 2 + 2√2 = 4.828+

12

s = 26/5 = 5.2

16

s = 28/5 = 5.6

20

s = 4(14+3√3)/13 = 5.906+

27

s = 6

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/24/12.