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SCOTT SIMON, host:

This is WEEKEND EDITION from NPR News. I'm Scott Simon.

And this week, our math guy, Keith Devlin, joins us to talk about the mathematics behind solving what can be a most irritating problem, a wobbly table. A new theorem explains that rotating a table will stabilize it.

He joins us from the studios of Stanford University.

Keith, thanks very much for being with us.

Professor KEITH DEVLIN (Stanford University): Hi, Scott. Thanks for having me on again.

SIMON: So how do you rotate the table without the plates falling off?

(Soundbite of laughter)

Prof. DEVLIN: You have to do it very carefully. There's a couple of caveats. Mathematicians don't know how to solve a wobbly table if it's wobbly because one of the legs is uneven. If one of the legs is shorter than the other, mathematicians are not much help to you. We're not much help to you also if the wobble is because the floor is actually stepped. You know, these lovely little outdoor restaurants in Tucson hill towns that are on the sides of hills?

SIMON: I've read about them, Keith. That's about the extent of my knowledge. Yeah?

(Soundbite of laughter)

Prof. DEVLIN: If the wobbliness is because you've got a stepped floor, mathematics doesn't help you either. Although if you're in a Tucson restaurant, who cares about the mathematics? If you have a perfectly made table and it's on a floor that's essentially smooth but wobbly, providing the wobbles are never more than 35-degrees - which is not much of a restriction, because at 35 degrees the pasta would slide off your plate anyway, then you can always correct the wobble by rotating the table by at most 90 degrees. And you're guaranteed to eliminate the wobble.

SIMON: What is the mathematics behind it?

Prof. DEVLIN: It's actually kind of neat, the original argument. You imagine you're looking at the table - it's a rectangular table or a square table. Three of the legs are going to be on the ground and one of the legs will be in the air. And let's imagine, as you look at it, the leg in top left is in the air. Suppose for a minute that the floor was made of sand. If you were to press the top left and top right hand corners, press them down evenly until the top left leg hits the ground, you will eliminate the wobble, but the top right leg will now push down into the sand, so it will be beneath the ground.

So now let's forget the sand and suppose we're back on the concrete floor. Suppose instead of pushing down, you rotate it clockwise. Initially, the top left hand corner leg is in the air. After the rotation, the leg in the top left has become the leg in the top right and it's beneath the ground. How can the leg at the top left that's above the ground get to be beneath the ground through rotating? At some point it must hit the ground. And at that point, since you're not on sand, it won't (unintelligible) stops.

So we know that somewhere between the starting point and the end point of that 90 degree rotation, at some point it must hit the ground. And when you reach that point, you've got rid of the wobble.

SIMON: Keith, was this something that waiters in restaurants knew but it took mathematicians hundreds of years to figure out?

(Soundbite of laughter)

Prof. DEVLIN: Yeah. You know, every waiter that I've known has done what I tended to have always done, which is get a napkin out, fold it up a few times and stick it under the wobbly leg. Mathematicians first noticed this back in '60s. A British mathematician observed that there's really a mathematical problem here. And some years later, in 1973, the very famous columnist Martin Gardner, in his Scientific American column, he gave a plausible argument that rotating would actually solve the problem. But it wasn't until last year that a group of four mathematicians finally resolved it. And it took a 15-page paper to fully explain the solutions.

SIMON: Keith, I have met your lovely wife. But I'm wondering if back in your dating days you would sit down for dinner date and the table would wobble a bit and you would say, I'll take care of that, and rotate the table and...

Prof. DEVLIN: I think the risk factor of spilling wine or whatever all over one's - the object of one's affection would be way too great. I think I would just sit there and put up with the wobble.

SIMON: Keith Devlin, who's executive director of the Center for the Study of Language and Information at Stanford University, his most recent book, by the way, is called "The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs)."

Keith, thanks so much.

Prof. DEVLIN: Okay, my pleasure, Scott.

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