The easiest way to look at it is not deduction but induction.
Start by noting down the properties of the 2D cube, then move on to the 3D cube.
A pattern should readily emerge.
Now, for a formal proof, yes there exists an Euler characteristic.
lets construct a 4D cube, it has 2^4=16 corners and 2*4=8 3D faces and 24 2 dimensional faces. the 8 3D faces contribute 4 edges and the total edges is 8*4=32. so can you know these properties by deduction or visualization somehow? maybe there is an Euler characteristic?
anyway I guess people are discussing fiber bundles and cotangent bundles here but I wanted to ask this
The easiest way to look at it is not deduction but induction.
Start by noting down the properties of the 2D cube, then move on to the 3D cube.
A pattern should readily emerge.
Now, for a formal proof, yes there exists an Euler characteristic.