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So it is easy to find number of edges given a half-square and half-cube by simply drawing, or just by intuition. We have two cases in both of these.

The square has 4 line of symmetry. If we define half-square by a line going through the edges, we can say half-square has 3 edges. If we define it by a line going through the sides, we can say half-square has 2 edges (or, well, 4). For a half-cube by the same logic we either have 6 or 4 (or 8).

The question is how do you generalize it to an n-cube? How do you even find how many different cases are you looking for? And even if you manage that, how can you determine how many edges does a half-n-cube has?

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    what is a half-hypercube? – 2017-02-15
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    It is what we get if we consider only one side of a hyperplae of symmetry,I guess. I'd look at it the same way as the half-square is either a right triangle (if the symmetry line we choose goes through the edges) or a rectangle (if the symmetry line we choose goes through the sides) – 2017-02-15
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    a half-square is a rectangle? – 2017-02-15
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    Like this: http://www.loisterms.com/onehalf.jpg . The other choice is this: https://debgeyer.files.wordpress.com/2008/11/halfsquaretriangle31.jpg . I'd take them both as half of a square – 2017-02-15

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