Hersh, Reuben - What Is Mathematics, Really?

This section has two purposes. It's a worked exercise in Pólya's heuristic (see Chapter 11).

At the same time, it's an inquiry into mathematical existence. By guided induction and intelligent guessing, you'll count the parts of a 4-dimensional cube. Then you'll be asked, "Does your work make sense? What kind of sense does it make?"

You're familiar with two-dimensional cubes (squares) and three-dimensional cubes. Is there a four-dimensional cube?