Some people claim to be able to visualize four dimensional objects. I have a friend who thinks that these folks are just blowing hot air. I thought it would be fun to test how well people actually can visualize 4D by asking them some questions about 4D objects which should be provable either by direct visualization or by more conventional means, for verification. Preferably they should not have easy combinatorial answers (e.g. "How many vertices does a 4-cube have?" should be excluded on this ground), so no one can bluff.
Here are some examples I came up with:
There are two types of "cylinder" in 4D: $D_3 \times [0,1]$ and $D_2 \times [0,1]^2$ (where $D_k$ is a k-dimensional ball). Consider $D_2\times [0,1]^2$. How many codimension-1 faces does it have? How about $S^1 \times [0,1]^2$?
(Answer: 5 for $D_2\times [0,1]^2$, 1 for $S^1\times [0,1]^2$. This one is marginal on the "easy combinatorial answer" criterion if you've studied any differential topology...)
Any knot can be unknotted in 4D, but some 2D surfaces can be "knotted" in a way that can't be undone. Is it possible to knot two Klein bottles in this way?
(I don't know the answer to this one.)
I don't claim these are the best examples, they're just presented to show the flavor of problem that I want. Ideally, please present solutions (or references) to any problems you suggest.