Swivel Chair Theorem

Can you always reorient a four legged chair to make it stable on an uneven surface?

If we can assume the feet form a perfect square, then: Yes!

Without loss of generality let's suppose that the chair starts off in the position shown top left, with its front right foot in the air and the others on the ground.  If we keep the two left feet on the ground while pushing down on the other two, the above ground foot will eventually touch the floor and the remaining foot will sink below it.

Now imagine instead rotating the chair by 90$^\circ$, keeping the same 3 feet touching the ground at all times, and ending with the two back feet in the positions previously occupied by the two left feet.  The two back feet and the front left foot occupy three corners of the square described by the feet at the end of the previous paragraph, so the fourth must be below ground.  Since the front right foot starts off in the air and ends up below ground we can infer from the intermediate value theorem that it must have touched the ground at some point.

This isn't much use with ordinary chairs because you end up pointing in a random direction.  However, it should work with swivel chairs and I've actually used it with four legged circular tables.  It makes a nice alternative to beer mat wedging!

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