= Solution
Label <vertices> by triples of bits and join pairs differing in exactly one bit. Draw the four <vertices> with first bit zero as an outer square, and the four with first bit one as an inner square, in the same cyclic order $00,01,11,10$ for the last two bits. Connect corresponding corners by four nonintersecting spokes. Every cube <edge> is present and there are no crossings, giving the required planar embedding.
\Image[/past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2008-planar-cube.png]
{title=Planar embedding of the cube graph with binary vertex labels}
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