A perfectly symmetrical square 4-legged table is standing in a room with a continuous but uneven floor. Is it always possible to position the table in such a way that it doesn't wobble, i.e. all four legs are touching the floor?
The answer is yes. Let the feet of the table clockwise are labeled with 1, 2, 3, 4 clockwise. Place the table in the room such that 3 of its feet - say 1, 2, 3, touch the ground. If foot 4 is on the ground, then the problem is solved. Otherwise it is easy to see that we can not put it there if we keep legs 2 and 3 on the same places. Now start rotating the table clockwise, keeping feet 1, 2 and 3 on the ground at all times. If at some point foot 4 touches the ground as well, the problem is solved. Otherwise continue rotating until foot 1 goes to the place where foot 2 was and foot 2 goes to the place where foot 3 was. Foot 3 will be on the ground, but this contradicts the observation that initially we couldn't place legs 2, 3 and 4 on the ground without replacing feet 2 and 3.