Seattle Themed Volume Problem

During the recent Joint Mathematics Meetings, which took place in Seattle, exhibitors were encouraged to contribute to the festive atmosphere of the Opening Night Reception by having a special activity or a contest. Following my long-standing tradition of having Geometiles structures in the theme of the conference host city, I decided to focus on coffee beans. After all, we were in the birthplace of Starbucks!

I wanted to give the participants of the contest an opportunity to flex their geometrical muscles, so I came up with the following problem.

The yellow and green tetrahedra are both completely filled with coffee beans. The yellow one contains 93 beans. How many coffee beans are in the green tetrahedron?

The green tetrahedron is regular, and the yellow tetrahedron faces are 3 right isosceles triangles and one equilateral triangle. The presence of the second tetrahedron is a hint. There were lots of special right triangles around the booth, and participants were encouraged to use them as tools

I had never run contest at a math conference before, and I was pleasantly surprised at how well it went. A number of math graduate students and postdocs stopped by, and I could just see the wheels turning. There was everything from recalling long lost volume formulas to just eyeballing.

Most of the participants guessed that the volume of the yellow tetrahedron is half that of the green one. Here is a “proof without words” of this fact.

The combined (idealized) volume of the two yellow tetrahedra equals the volume of the green tetrahedron by Cavalieri’s principle. In the picture below, the two yellow tetrahedra have been made into one.

The yellow tetrahedron has the same height as the green one, and its cross section at any given height is an equilateral triangle identical to the cross section of the green one. So by Cavalieri’s principle, the two tetrahedra have the same volume. We have a whole activity book based on Volumes and Cavalieri’s Principle— check it out!

Cavalieri’s principle is actually a bit of overkill for this particular problem— one can do everything with similar tetrahedra due to the linear nature of the shapes. However, with Cavalieri’s principle one need not think of any formulas at all; this is a helpful feature when you are solving a problem in an exhibit hall!

Here’s a video illustrating the application of Cavalieri’s principle to this problem. The idea can easily be generalized to other shapes.

Back to the original problem, the number of beans in the green tetrahedron had to be greater than twice 93. That’s because coffee beans have finite volume, and they can’t get into the crevices formed by some of the smaller dihedral angles of the smaller yellow tetrahedra. Also, in the green tetrahedron there is just more room for the beans to optimize their closeness to one another. All in all, the green tetrahedron contained 204 beans.

There is definitely an element of luck and randomness here. The person who got the closest to the right number, with a guess of 205, was David Weed, a first year graduate student in mathematics at UC Riverside. Here is David pictured with the Geometiles Mini 3 set he won:

Kaiying O’Hare, a postdoc at Johns Hopkins University, was a close second with 208.

I look forward to trying variations of this contest on other audiences at future events!