Wobbling Table

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?


SOLUTION

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.