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    $\begingroup$ Great stories! Related MO links: Gerver's sofa. Sofa in a snaky 3D corridor. $\endgroup$ Commented Apr 15, 2020 at 22:32
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    $\begingroup$ The Conway car problem is closely related to the ambidextrous sofa problem mentioned in the Wikipedia link $\endgroup$ Commented Apr 16, 2020 at 19:05