You’re standing in a room with an uneven floor. Before you is a square table with four legs. The table wobbles, but by turning it gradually you manage to find a position in which all four feet are supported, eliminating the wobble (though now the tabletop isn’t level).
You wonder: Is this always possible? Assuming that the four legs are of equal length and that the surface of the floor varies smoothly, is it always possible to position a four-legged table so that all four legs are supported?