Maths · the intermediate value theorem at the café
Your café table rocks. Before you fold a napkin under a leg, try this: turn the whole table on the spot. On most floors, somewhere within a quarter-turn, all four legs land at once.
Any three points in space lie on a flat plane. That is why a three-legged stool never rocks, whatever floor you put it on: its three feet always find the ground. A four-legged table is different. Its feet sit in one fixed plane, and a bumpy floor usually doesn't offer four points in a plane in exactly the right places. So the table rests on three legs and the fourth hangs a few millimetres in the air. Lean on the wrong corner and it clunks onto the fourth leg while another one lifts.
A wobbling square table always rocks across a diagonal. Picture the two diagonals: orange legs at two opposite corners and blue legs at the other two. If the floor under the orange pair is, in total, a bit higher than under the blue pair, the table balances on the orange diagonal like a seesaw and tips toward one blue leg or the other.
Now turn the table about its centre and keep track of one number: how much higher the floor is under the orange pair than under the blue pair. Call it the mismatch. When it is zero, all four feet can touch at once. When it is positive, orange is high; when negative, blue is high.
Here is the trick. Turn a square table by exactly 90° and every leg ends up where its neighbour used to be. The orange legs now stand exactly where the blue ones stood, and vice versa. So the mismatch after a quarter-turn is precisely the negative of the mismatch you started with. If you began orange-high by 3 mm, you finish blue-high by 3 mm.
A quantity that changes smoothly from +3 to −3 has to pass through 0 somewhere in between.
That is the intermediate value theorem, one of the first results in any calculus course, and it is doing real work here. As long as the floor has no sudden steps, the mismatch changes continuously as you turn. It cannot jump from positive to negative without hitting zero. At that angle the table stands on all four legs. Drag the table in the picture above and watch the curve: however you reshuffle the floor, the first quarter-turn always contains a crossing.
Martin Gardner described the rotating-table argument in his Mathematical Games column in Scientific American in May 1973. For decades it circulated as folklore, and the folklore version quietly assumes things. In 2005 Bill Baritompa, Rainer Löwen, Burkard Polster and Marty Ross posted a careful treatment, Mathematical Table Turning Revisited, and André Martin published a related analysis in Physics Letters A in 2007. The conditions that emerged:
And one catch that the theorem never promises to fix: steady is not the same as level. On a sloping terrace the table will stop rocking, but your coffee may still sit at a slight tilt.
The same kind of argument turns up in surprising places. Walk once around the equator and compare the temperature at each spot with the temperature on the exact opposite side of the Earth. The difference flips sign halfway round, so somewhere there must be two opposite points with exactly the same temperature. No measurement needed, only the fact that temperature changes smoothly.
So next time a table rocks, don't reach for the sugar packets. Grip it by two opposite corners, turn slowly, and feel for the moment the wobble dies. Somewhere in that quarter-turn, the maths says it has to.