Some open problems

The problems here were chosen because they can be stated in a simple and elementary way. I don't go into references to results concerning them, but if you intend to think about them seriously I encourage you to check the literature !

The angel and the devil.

Unfolding polyhedra: given a convex polyhedron (in Euclidean 3-space), is it possible to cut it along some of its edges so that the resulting surface is connected and can be unfolded on a plane, without self-intersection ?

The Kneser-Poulsen conjecture: given $N$ balls in Euclidean $n$-space, suppose one changes their position so that the distance between the centers of any two balls is greater in the new configuration. Is the volume of the intersection of the balls then smaller than in the first configuration ? Is the volume of the union of the balls greater ?

Is there a triangulation of the torus with all vertices of degree 6 except one of degree 5 and one of degree 7 ? I heard this problem from Ken Stephenson, who said it was open.

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