# 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?

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.

 2s = 2 3s = 2 + 8/√65 = 2.992+ (DC) 6s = 3 7s = 18/5 = 3.6 (DC) (MM)

 8s = 3.683+ (MM) 9s = 3.890+ (DC) 10s = 3.985+ 16s = 4

 18s = 2(7+√19)/5 = 4.543+ (MM) 19s = 4.709+ (DC) 20s = 4.816+ (DC) 21s = 2 + 2√2 = 4.828+

 22s = 4.947+ (DC) 24s = 4.959+ (DC) 30s = 5 31s = (5+√31)/2 = 5.283+

 32s = 2 + 12/√13 = 5.328+ 33s = 5.512+ 34s = 5.546+ 35s = 5.669+ (DC)

 36s = 5.714+ (DC) 37s = 4 + √3 = 5.732+ 40s = 2 + √2 + √6 = 5.863+ 41s = 4(14+3√3)/13 = 5.906+

 42s = 5.970+ (DC) 44s = 5.988+ (DC) 46s = 5.999+ (DC) 54s = 6

Here are the best-known packings of soup cans.

 1s = 2 2s = 2 + √2 = 3.414+ 3s = 3 + 1/√2 = 3.707+ 8s = 4

 10s = 2 + 2√2 = 4.828+ 12s = 26/5 = 5.2 16s = 28/5 = 5.6 20s = 4(14+3√3)/13 = 5.906+
 27s = 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.