Problem 26: Getting Hyper with Cubes

An n-dimensional unit hypercube is a shape whose vertices are all the n-dimensional points whose coordinates are either 0 or 1.

For example the 2-dimensional unit hypercube with vertices at (0,0), (1,0), (0,1) and (1,1) is a square. The square has 4 vertices and 4 edges.

The 3-dimensional unit hypercube with vertices at (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1) is a cube.

The 4-dimensional unit hypercube with vertices at (0,0,0,0), (1,0,0,0), …, (0,1,1,1), (1,1,1,1) is a tesseract.

1. How many vertices and edges does the unit cube have?
2. How many vertices and edges does the unit tesseract have?
3. Can you come up with a method to work out the number of vertices and the number of edges for any n-dimensional unit hypercube?