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.