imc

For simplicity, we will name this cube $ABCDEFGH$ by vertices, as shown below. Note that for each face of this cube, two edges are labeled $0,$ and two edges are labeled $1.$ For all twelve edges of this cube, we conclude that six edges are labeled $0,$ and six edges are labeled $1.$ We apply casework to face $ABCD.$ Recall that there are $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$ ways to label its edges: 1. Opposite edges have the same label. There are $2$ ways to label the edges of $ABCD.$ We will consider one of the ways, then multiply the count by $2.$ Without loss of generality, we assume that $\overline{AB},\overline{BC},\overline{CD},\overline{DA}$ are labeled $1,0,1,0,$ respectively: We apply casework to the label of $\overline{AE},$ as shown below. We have $2\cdot 2=4$ such labelings for this case. 4. Opposite edges have different labels. There are $4$ ways to label the edges of $ABCD.$ We will consider one of the ways, then multiply the count by $4.$ Without loss of generality, we assume that $\overline{AB},\overline{BC},\overline{CD},\overline{DA}$ are labeled $1,1,0,0,$ respectively: We apply casework to the labels of $\overline{AE}$ and $\overline{BF},$ as shown below. We have $4\cdot 4=16$ such labelings for this case. Therefore, we have $4+16=\boxed{20}$ such labelings in total.