I worked at some problem with moving pipe around a corner and i was trying to fing the maximum length of pipe for corner with corridor lengths 1. It is pretty obvious that maximum length is √2, since that is the length of pipe when it's in position, when the angles where it touches the outer walls are the same - 45 degree.
It's obvious that if pipe can get through this point, it can move around the corner. But i don't know how to exactly mathematically prove it.
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