Macadamia Grocery Geometry Challenge

For the Museum of Mathematics

Another year, and another visit from MoMath’s Hawaiian colleague, Dave Masunaga. This year, the macadamia nuts came in six cylinders, instead of many tetrahedra.

Somehow the number six reminded us of the classic challenge to make six pencils all touch each other simultaneously. However, the classic solution, which involves two layers roughly like the center three in the picture below, does not work with macadamia cylinders because they are too short compared to their diameter.

So that begged the question: what’s the maximum number of these cylinders which can simultaneously mutually touch? Three is easy:

And we found various configurations for four:

But five proved elusive. These cylinders were 9.5 centimeters high and 7 centimeters in diameter. So here’s the challenge: see if you can find a way to have five (or more?) cylinders of an aspect ratio at least close to 9.5 : 7 all touch each other simultaneously, and send a picture of the configuration to mondays@momath.org — the first/best solutions will be posted in a future installment. Happy Grocery Geometry!