How many distinct ways are there to label the faces of a cube with distinct numbers from 1 to 6, such that there is at least one pair of opposite faces which do not sum to 7?

Question

Rotations (which preserve orientation) are considered the same way.
Reflections (which do not preserve orientation) are considered distinct.

Answer 1

Without loss of generality, let the bottom face be one.
Then there are five choices for the top face and six choices for the other sides.
Of these, exactly one of them has faces which all sum to seven, hence the answer is $5 \cdot 6 - 1 = 29.$