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?
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Yes, it is. Label the legs A, B, C, and D (going clockwise), and position the table so that leg A is up in the air and B, C, and D are touching the floor. Call this Position 1. Now imagine that the floor is made of sand. Leaving C and D on the surface, push B into the sand until A touches the floor. Call this Position 2. Now return to Position 1 and, instead of pushing down, rotate the table clockwise, keeping legs B, C, and D in contact with the floor. Because the legs are of equal length, we know that, as you turn the table through 90 degrees toward Position 2, leg A will at some point pass beneath the surface of the floor. So there must be some intermediate point at which all four legs just touch the floor, and the table is stable.
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