Answers to the second set of 4D geometry practice problems:

1.Two ways to get the answer:

Reason it out by thinking about the bases and connecting edges:

Each base has 6 vertices, so with two bases there are 12 vertices.


Each base has 9 edges, so with two bases there are 18 blue edges, and the base has 6 vertices so there will be 6 connecting edges; the prism will have a total of 18+6=24 edges.

If you're a really good artist, you can also draw it out and count (but bring colored pencils)

Because each vertex of the 3D shape has 3 edges connected to it, you could also use Brian's formula: 6*4=24 edges

2: First note that the 3D shape has 6 vertices and 12 edges
There are 6*2=12 vertices, and 12*2+6=30 edges

Here's a picture to help me explain my answers:

(notice that I drew the two bases exactly the same in one color, and then connected the vertices using another color to help me keep track of what I was doing)

A. Explain why the number of vertices in the 4D prism is twice the number of vertices on the base:

All of the vertices are either on the top base or the bottom base. There are 6 vertices on each base, and two identical bases, so there are 6*2 bases total

B. Explain why the number of edges in the 4D prism is twice the number of edges on the base plus the number of vertices on the base.

There are 12 edges on each base, and there are two bases, so that makes 2*12=24 edges. Then there are edges connecting the vertices of the "top" base to the vertices of the "bottom" base--one edge (straw) for each of the "bottom" vertices. That makes 6 more edges: 24+6=30 edges.

Note that in this example, each vertex of the 3D shape has 4 edges connected to it, so you could use reasoning like Brian's to deduce that there were 6*5 edges in the 4D shape.

3. Alice is figuring out the number of vertices and edges in the 4D prism with base:

A. Alice knows that the 4D prism will have more vertices than the base, which has 5 vertices. How many more vertices will there be on a 4D prism? Why?

A 4D prism will have 5+_5___ vertices because... there are 5 vertices on the "top" base, and 5 more on the "bottom" base.

B. Alice knows that the 4D prism will have more edges than there are on the base, which has 8 edges. How many more edges will there be on a 4D prism? Why?

A 4D prism will have 8+_8__+_5__ vertices because... there are 8 edges on the "bottom" base, 8 edges on the "top" base, and 5 edges that connect the 5 vertices on the "bottom" base to the 5 vertices on the "top" base.


Note that different vertices on this shape have different numbers of edges connected to them, so you can't use Brian's reasoning to solve this one at all...