A: To get to the same side!
(old math joke)
In 1865, August Ferdinand Möbius published his paper "Über die Bestimmung des Inhaltes eines Polyëders" ("On the determination of the interior of some Polyhedra"), in which he introduced the one-sided surface that nowadays is called, after him, "Möbius strip".
In the picture, we can see the description of the surface originally given by Möbius: it is the usual identification space of the rectangle that one studies in the first Topology course.
References
A. F. Möbius: Über die Bestimmung des Inhaltes eines Polyëders, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften, Mathematisch-physikalische Klasse, Bd. 17, p. 31-68, 1865.
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