Answers

1. How do you make a 4D cube (or make a 3D shadow of a 4D cube)?

Take two 3D cubes and add edges to connect their vertices to each other. More specifically, you add an edge at each vertex of one of the 3D cubes that ends at the corresponding vertex of the other 3D cube.

2. How many vertices does a 4D cube have?

Hmmm...I'll have to come back to this one. OK, I finished problem 3. Now I know there are 16 vertices.

3. Tell how you can use the number of vertices a 3D cube to find the number of vertices on a 4D cube, and explain why that strategy makes sense.

Answer #1: One 3D cube has 8 vertices. You make a 4D cube out of 2 3D cubes, so that makes 2*8=16 vertices. When you connect the 3D cubes, you don't add any new vertices, so you just have the 16 vertices from the 2 3D cubes.

Answer #2: What you do, is you double the number of vertices that a 3D cube has, and that makes sense because you make a 4D cube out of two 3D cubes, and then add some connecting edges, but no more vertices, so you just get the vertices from the two 3D cubes.

4. How many edges does a 4D cube have?

I clearly should have put problem 2 after 3 and 4 after 5.... OK, I came back. 32 edges.

5. Tell how you can use the number of edges and vertices on a 3D cube to find the number of edges on a 4D cube, and explain why that strategy makes sense.

Answer #1: One 3D cube has 12 edges. You make a 4D cube out of two 3D cubes, so that makes 2*12=24 edges, just from those 3D cubes. After you have the two 3D cubes, you connect the vertices with more edges, and if you look at just one of the 3D cubes, there's a connecting edge coming from each vertex. A 3D cube has 8 vertices, so that makes 8 more edges, for a total of 24+8=32 edges.

Answer #2: What you do is that you double the number of edges in a 3D cube and then add the number of vertices in a 3D cube. This makes sense because you have two 3D cubes making your 4D cube, and those 3D cubes together have 2* the number of edges in one 3D cube. When you connect the 3D cubes, you get a connecting edge for each vertex on one of the 3D cubes, so if you add the number of vertices in a 3D cube, you get the total number of edges in a 4D cube.

Note: for #3 and 5, you should know and be able to write an answer like either answer #1 or answer #2, your choice.