Boarders on Nostr: - A 0-dim cube has 1 0-dim face (vertex) and nothing else. Total: 1 - A 1-dim cube ...

- A 0-dim cube has 1 0-dim face (vertex) and nothing else. Total: 1

- A 1-dim cube cube has 2 0-dim faces and 1 1-dim face (edge). Total: 3

- A 2-dim cube has 4 0-dim faces, 4 1-dim faces and 1 2-dim face. Total: 9

- A 3-dim cube has 8 0-dim faces, 12 1-dim faces, 6 2-dim faces and 1 3-dim face. Total: 27

Puzzles:
1. What is the pattern for the 4-cube?
2. What is the total for the n-cube?
3. Why is that the total?
4. What is the general formula for the number of k-dim faces of the n-cube?
[from “What is Mathematics, Really?” by Hersh]