A wobbly table seems to call for a wedge, a folded napkin, or a different spot. But sometimes simply rotating it is enough: the irregular floor forces a balanced position to exist.
The Wobbly Table
Riddle statement
A square table has four equal legs whose endpoints form a square.
It stands on a continuous, smoothly irregular floor: there are no steps or sudden jumps, and during the rotation three legs can be kept touching the floor.
The table wobbles in its current orientation. Without wedges and without moving it to another place, can we guarantee that some orientation obtained by rotating it around its center will make all four legs touch the floor?
Show solution
Solution
Yes, under the stated conditions.
While three legs remain touching the floor, measure the gap between the fourth leg and the floor. We can assign a sign to this quantity depending on which of the two diagonal legs is above its support position.
Now rotate the table slowly around its center. After a 90-degree rotation, each leg occupies the position previously occupied by the next one. The diagonal that was high and the one that was low exchange roles, so the support difference changes sign.
Since the floor and the motion are continuous, that difference also changes continuously. It cannot go from positive to negative without being zero at some instant.
At that instant, the fourth leg touches the floor at the same time as the other three.
The conclusion depends on the hypotheses in the statement. A step, a sudden hole, or ground so uneven that it prevents three legs from staying in contact would break the argument.
Answer: yes. On a continuous, smoothly irregular floor, there is a stable orientation.