Sun 28 Dec 2008

## The “Shaky Table” puzzle

Posted by Happy under puzzle

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A mathematician goes into a restaurant and is seated at a square table, which he notices is unstable. The table has four legs evenly spaced, but although the floor is smooth he can see it is quite uneven. As a result one leg of the table does not quite reach the floor.

The waiter offers to wedge a piece of folded paper under the table leg, but the mathematician quickly analyses the problem and comes up with a better solution. Without getting up from his chair, he quickly moves the table so that it is now stable. Can you find how he did it?

The puzzle is to prove that there is a way to move the table to a nearby location so that all 4 legs rest evenly on the floor, and to find a simple method to do so.

You can assume that there are no holes, obstructions or other artificial restrictions. The solution should be perfectly practical under normal conditions, and may actually turn out to be rather useful!

__Solution__

Rotate the table around its centre axis in either direction. A stable location is guaranteed to be found with less than a 90 degree rotation.

__Explanation__

The table starts with 3 legs (A,B,C) on the floor and one (D) above it. If a clockwise rotation of 90 degrees were completed then each leg would move into the position of its neighbour in that direction. At some intermediate point C lifts off the floor (on its way to replace D) and a point where D strikes the floor (on its way to replace A). The technique used ensures there is no point where two legs are off the floor, therefore these two points are reached at coincident or overlapping rotations. There is thus a point or region of stability where all four legs are on the floor.

The logic breaks down if the floor has discontinuities or holes. On a smooth floor it always works.