Same crust length both ways, more naked one of the ways...
It shouldn't be the same crust length. If you cut straight across, from the middle of one side to the middle of the opposite side, then the perimeter should add up to 3s. That is, if the length of each side is s (assuming a square sandwich), then the perimeter, p, of each half-sandwich is p = s + s + 1/2-s + 1/2-s = 3s. On the other hand, if one cuts diagonally, from corner to opposite corner, then the p of each half-sandwich is p = s + s + (s * sqrt2) = 2s + (1.4142 * s) = 3.4142s.
This can be generalized to the more general situation where the full sandwich is a rectangle where the length is not equal to the width.
However, each half-sandwich should still have the same upper surface area - for the straight cut, area of each half-sandwich = s * (1/2 * s) = (s^2) / 2, and for the diagonal cut, the area of each half-sandwich = 1/2 * (s * s) = (s^2) / 2.
Finally, since each half-sandwich will have the same upper surface area, and by definition should have the same height, each half-sandwich will have the same volume = h * ((s^2) / 2), and therefore each half-sandwich will contain the same "amount" of sandwich - bread plus fillings - no matter how it's sliced.
QED
