A deleted Mathoverflow post - stored

A post at mathoverflow can sometimes get deleted by a bot. The following was: https://mathoverflow.net/questions/416530/thinnest-3-fold-and-n-fold-coverings-of-the-plane-by-congruent-convex-shapes. So, saving a copy here.
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This post is totally based on https://mathoverflow.net/questions/183069/thinnest-2-fold-coverings-of-the-plane-by-congruent-convex-shapes (Thinnest 2-fold coverings of the plane by congruent convex shapes) - indeed, am only recording a couple of very natural further questions here.

Are there convex planar regions which do not tile the plane and do not form a perfect 2-cover of the plane but do form a perfect 3-cover (generalization from 3 to n is natural)?

In 3D, are there convex regions that cannot tile but can form a perfect 2-cover of 3-space?
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Note:This post linked above features a superb demonstration by Noam Elkies.