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I first time I heard about the Moving Sofa Problem was in High School and I didn't understand why it was hard. I thought that if you assumed everything to be the initial sofa and then cut away all the pieces that exited the hallway, you would easily get the maximum area.
Not much later I realized that the problem resided on how you perform that movement. The state space is the space of all trajectories (including direction) that make the turn. The problem is hard because this is a infinite dimensional state space and the functional that defines the area given the curve and direction is not well-behaved at all.
Nonetheless, the initial realization which formally says that the state space doesn't need to be viewed as the space of shapes of sofas (which seems very intractable), but as the space of trajectories (which is also intractable but less scary) is nice to animate.
Many years later, with the initial aim of making a expository interactive blog post for SoME2, in a train trip from Madrid to Barcelona I coded the following interactive animation. I never ended up having time to write the blog post.
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