That annoying restaurant table that rocks back and forth? Mathematics guarantees you can fix it by just rotating it—no napkins needed! Rotate by at most 90° and the table will be stable.
When w(θ) crosses zero, the table is stable!
Imagine a square table with 4 equal-length legs on an uneven floor. At any position, 3 legs touch the ground while 1 leg floats (or 2 diagonal legs float while the other 2 touch).
Define a "wobble function" w(θ) = height of floating diagonal pair minus height of touching diagonal pair. As you rotate, this function changes continuously.
At θ=0°, suppose w(0) > 0 (one diagonal floats). At θ=90°, the table is in the same position but rotated, so w(90°) = −w(0) < 0. By the Intermediate Value Theorem, w(θ) = 0 somewhere in between!
Mathematics guarantees a stable position exists within 90° of rotation. This was rigorously proved in 2005 by Baritompa, Löwen, Polster, and Ross, though the idea dates to the 1960s.