Igor Pak has just published a preprint on a cute variant of the teabag problem on the maximum volume of a surface enclosed by two squares joined together at their edges and warped without stretching. Ordinary paper provides a good model of a surface that can be bent and folded but not stretched, so this problem is easy to make physical models for, although the volume calculations or measurements may be difficult.
Anyway, in Pak's version, he considers a different surface, formed by six squares joined together at their edges: the surface of a unit cube. Of course, these six squares can enclose a cubical region with unit volume. And it's easy to dent parts of the cube inwards, to enclose smaller volume. What Pak shows, perhaps surprisingly, is that unit volume is not optimal: it's possible to bend and fold the faces of a cube, without stretching them, to enclose larger than unit volume. Pak also cites several earlier works on similar problems of which I was unaware.
PS re a different paper also appearing on arXiv tonight: telling me your papers are "not mistake free" is not a good way to persuade me to read them.