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  1. How many corner does a cube have in 4 dimensions? How many 3D faces?

Now by observation we can tell that, an n-dimensional cube has $2^n$ corners. 

  • 1D cube which is a line have $2^1$ corners
  • 2D cube which is a square have $2^2$ corners
  • 3D cube have $2^3$ corners

Hence, 4D cube have $16$ corners.

There are 16 corners and degree of every corner is 4, from this we can find out the number of 1-cubes,

Number of 1-cubes (line) in 4 dimensional cube = $\large \frac{16\times 4 }{2} = 32$ 

For calculating number of 2-cubes,

  • Every line in a square (2-cube) participates in forming only 1 square (2-cube).
  • Every line in a cube (3-cube) participates in forming only 2 squares (2-cube).

and Every line in a hypercube (4-cube) participates in forming 3 squares (2-cube).

 

Every line is participating in forming 3 squares (2-cube) and there are totally 32 lines.

 

Total number of 2-cubes or squares in a 4 dimensional cube = $\large \frac{32 \times 3}{4} $$=24$

Now the last question is Number of 3-cube in a 4-cube.

By observation we can see that every N-cube have $|2n|$ cubes of dimension (N-1).

  • a line has 2 zero dimensional points.
  • a square have 4 one-dimensional lines
  • a cube have 6 two-dimensional planes
  • a hypercube have 8 three-dimension cubes.

but this is the question i’m not able to answer. How every N-cube have $|2n|$ cubes of dimension (N-1)?

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