This is an image which caused my students a great deal of trouble. I thought it was pretty fun because I don’t think there is any cute trick counting this. I want to know how many faces, edges and vertices there are. If you are an advanced math lover then consider Euler’s equation relating to polyhedrons. Euler says:

number of edges = number of faces + number of vertices – 2

Does this equation hold here?

(If you getstuck on the 2nd question, there is a hint in the image mouse-over.)

##
About Samantha from SocialMath

Applied Mathematician and writer of socialmathematics.net.

Question: Is this image depicting the union of a small cube with two rectangular prisms, such that only edges and vertices are in common, or is this image depicting the union of a tall-and-thin rectangular prism with two flatter rectangular prism, so that the unseen volume is convex? The f-vectors for those two objects will be different.

Also, generalized Euler holds for graphs on surfaces, so which object is intended makes a difference as to the embedding surface. (That’s my fancy way of saying that I doubt the mouseover hint is necessarily relevant.)

Good question! The image is intended to be one polyhedron- so the “back” edges share only 1 vertical edge at the center back of the shape. I crafted a back view and added that to the post.

The Euler’s rule that we were working in class is quite basic having to do with convex vs concave polyhedrons which is why I felt the hint relevant. I actually don’t know very much about generalizing this equation for different surfaces. If I said we are viewing this polyhedron in R^3 or projecting onto R^2 would that make a difference?

(For the casual-mathematical folks there is a great AMS article about Euler’s Polyhedral Formula by John Malkevitch that may give you an idea of what sarah-marie is commenting on. Certainly, I learned some things I never knew about Euler’s theorem and graph theory.)