Some remarks: I mentioned in Post #6 that Edwin Abbot described, in Flatland, how to extend a cube of n dimensions to n+1. The method he gave is exactly as @Chronos describes just above. (And exactly as I would have described it.) But note something more: Note the recursive evolution of all these components. As each cube is extended to the next higher dimension, each of its components creates the new components of the next cube in a very predictable manner.
The Wiki page linked by @Gyrate in Post #2 has one diagram that illustrates this process also.
Here’s your next exercise: You can make a table showing all these numbers. On row 0, put the numbers for a “0-cube”, that is, just the number 1 for the 1 point. On row 1, put the numbers for the “1-cube” (just a line segment), that is, 1, 2 for the 1 line segment as a whole and the 2 endpoints.
On row 2, put the numbers for the “2-cube” (the square), that is, 1, 4, 4 for the 1 square as a whole, the 4 edges, and the 4 vertices.
On row 3, put the numbers for the “3-cube” (a real normal 3-D cube):
1, 6, 12, 8 for the 1 cube as a whole, the 6 faces, the 12 edges, and the 8 vertices.
On row 4, put the numbers for the “4-cube” (the tesseract), as given by @Chronos above: 1, 8, 24, 32, 16
What would rows 5 through 10 look like (for 5-D through 10-D cubes)? Just take each row of this chart and “evolve” the next row according the the pattern that @Chronos described.
THEN: When you’ve done that (okay, you don’t really have to do the entire 10 rows), and ONLY then, read on to my next post . . .