This puzzle celebrates our 20 puzzle point solver - Wang. It goes like this:

Wang and a friend have been shopping for new kitchen chairs. They went to a few stores, tried sitting on the wooden chairs, plastic chairs, aluminum chairs, counter stools and bar stools. To their surprise they found most of the four-legged chairs less stable than most of the three-legged chairs that they sat on. Wang has been contemplating about this all the way home and suddenly it occur ed to him that there is mathematical truth behind their empirical observation. What is it?

Wang and a friend have been shopping for new kitchen chairs. They went to a few stores, tried sitting on the wooden chairs, plastic chairs, aluminum chairs, counter stools and bar stools. To their surprise they found most of the four-legged chairs less stable than most of the three-legged chairs that they sat on. Wang has been contemplating about this all the way home and suddenly it occur ed to him that there is mathematical truth behind their empirical observation. What is it?

Top image by Fernando G., distributed under CCL.

Answers accepted all day long on Friday, on our Family Puzzle Marathon. They will be hidden until Saturday morning (EST) and everyone who contributed something reasonable will get a puzzle point. Please, explain your answer.

## 25 comments:

Three leg tables don't wobble even if all the legs aren't exactly the same which isn't true of having more legs. Two "legs" would allow movement in a straight arc. Three "legs" prevent the top from moving in any direction. Adding another leg gives the stool "choice" - it can balance on legs 1,2, and 3 leaving a wobble in 4 or any variation thereof. I hope I explained this clearly enough. I can piucture it but am not sure my language is visual enough.

Any three points can and will form a triangle, and all triangles will "lay flat" on a plane, on SOME plane. (In fact we could say all triangles are flat, and any triangle will always determine a plane.) The 3 legs of the chair might be uneven lengths, up to a point, and there will still be considerable stability.

It's much more challenging to make a 4-legged chair whose fourth point (of contact) is in the same plane as the first 3 points.

Any 3-legged stool or chair will be relatively stable. Even on an uneven floor. The three contact points will determine a plane in any case, although it might not be as horizontal as it feels.

In fact on some uneven floors it may be quite impossible for a 4-legged chair, or table, to be stable if it gets moved about. Five or six would be even worse.

Well, duh! Three points determine a plane, ergo a 3-legged stool/chair should never wobble!

Barry Gerard

It has to do with degrees of freedom.

With a one leg stool/chair, the top (seat) can move freely in two dimensions. When you add the second leg, you constrain one dimension, so the seat can only move freely in one dimension. When you add the third, it's constrained in all three dimensions (three points also define a plane).

So why is the fourth leg bad? Now the stool/chair can choose any three legs to make a plane - four different combinations! If there are any variations in the leg lengths or the floor, the chair will "rest" on a different plane based on the distribution of the weight of the person sitting on it. So, it will likely seem less stable!

3 points make a plane

A plane is defined by three points. One plane can contain three non collinear points.

A four legged chair has a possibility of four sets of three legs so four possible planes where the contact could take place. If the fourth leg does not have the same length than the other leggs then it will not be in the same plane and the chair will wobble being more unstable than a three legged chair.

Any 3 non-collinear points define exactly one plane. So, any 3 points (like the end points of 3 legs of a 3-legged stool) are coplanar and would all touch a plane (like a floor) at the same time.

4 points (like the end points of the legs of a 4-legged chair) are not guaranteed to all fit in one plane together. Often, four-legged pieces of furniture will wobble because of this. Even the best-made pieces of 4-legged furniture may wobble on your floor because your floor isn't likely to be uniformly level!

heheheh do you mean Tom? =)

From geometry we know that three points determine a plane -- the three points are the ends of the legs of the chair -- and they sit on the plane of the floor.

Four points could be co-planar, but that is unlikely and the chair will rock a bit.

Wow, I'm not sure I have the technical explanation here, but I'll try. Three points define a plane. So with three legs, the three ends of each leg determine one and only one plane. So the chair has nowhere to go. With four legs, you've determined multiple planes (four different ones), so basically your chair can choose between them. Most likely, two diagonal legs lift the chair slightly, and the chair wobbles between the two planes that include those two points.

Now to go fix my kitchen table, which has four legs and oscillates between planes as I type...

Yikes! That "Sophia" comment was actually from "Bean". We had a mix up on who was logged into the laptop.

So, three points form a plane, a chair with three legs determines a single plane, a chair with four determines multiple plane, it wobbles....

I'll get this right yet!

Since 3 points define a plane the three legs of the stool are always on the same plane. A chair with 4 legs will wobble if the floor is not even because 3 of the four legs will be on the same plane but the 4th will be on a different one making it unsteady.

A three-legged chair will be more stable than a four-legged chair because a three-legged chair has less legs, therefore less opportunities to have an uneven leg.

Since any three points form a plane, the three legged chairs "feet" would all lie on the same plane, making the chair or stool sturdy and not wobbly like a 4 legged chair with one leg shorter than the others.

Technically, I should've said any three NONCOLLINEAR points form a plane. oops!

Hi

First time poster. Love this blog, Maria.

Ok, so let's assume these stools are on some planet with a constant source of gravity. (Sorry, I'm an engineering prof.) ^_^

Let's say there were two legs and the bottom of each leg comes to a point (or close enough for our purposes). The two points describe a line, but the chair can wobble on either side of the line (just like a bicycle). So the line is fixed, but the stool can rotate as much as it wants around it.

Let's say there are three legs; three legs define a plane, and assuming that plane is horizontal to gravity (i.e., the floor) the stool will be stable. In other words, the degree of freedom of the two legged stool is constrained by the thrid leg. There is no room for movement.

Let's say there are four legs, and that's it's not possible to get all four legs exactly the same length. The two kitty-corner legs with the longest average length define the two legged stool, but we have once again introduced a degree of freedom; by shifting our weight the stool will rest on one, but not both of the other two legs. It wobbles between them.

Let's take this one step further to why office chairs have 5 legs (or wheels). Let's say a 3-legged stool is 3' high, and we put a 150# person on it. While sitting on the stool, it is realatively easy for that person to extend their center of gravity across the line which defines the two legged stool, away from the 3rd leg, and they are on their way to the floor. It is harder to get your center of gravity outside of a square (4 legs) and harder still to get outside of a pentagon (5 wheels).

The 3 legged stools are sturdier because all three feet must be on the same plane but a 4 legged stool could have the fourth foot on a separate plane and all the feet need to be on the same plane for it to be stable.

The answer has to do with 'degrees of freedom' – and the fact that we live in a 3-dimensional world.

Think of it this way:

(1) If you hold a one legged stool in the air, you can move it in any direction, twirl it, and so on. Its motion isn't constrained at all. That is, the top of the tool can move freely in 3 dimensions.

(2) If you put (and keep) the one leg on the ground, now its motion is constrained: you can't lift it, or rotate it... although you can swing the top around in a variety of different arcs. That is, the top of the cane can move freely in 2 dimensions.

(3) If you have a 2 legged stool and place the legs on the ground, you can still move the tops, but only along a single (straight) arc, back and forth. That is, the top can move freely in 1 dimension.

(4) If you try the same trick with a three legged stool, you can't move the top at all. The stool can move in zero dimensions...which is to say, they can't.

Each time you add a leg, you remove one dimension in which the top can move freely - that is, each new leg removes one 'degree of freedom'.

Now, what happens when you add a fourth leg? Well, now you have too many constraints. This means that there are multiple ways that the stool can 'solve' the problem of which legs to use for support. Wobbling occurs when the stool can't 'decide' which solution to use, or, more precisely, when it's changing its mind about which solution to use. In effect, during the time that the stool is actually wobbling, it's really a two-legged stool, with one degree of freedom - which is the direction of the wobble.

This behavior can be seen in systems of equations. If I have two variables, one equation will give me a whole range of solutions:

y = 2x + 4 Solutions include (1,6), (2,8), (3,10). In fact, the range of solutions is just the graph of the equation.

If I add another equation: y = 2x + 4 and y=3x-1 there is only one solution (5,14)

That is, each new equation removes a 'degree of freedom' from the system. With two variables, I need two equations to lock things down. With three variables, I need three equations. And so on.

Now what if I add a third equation? y = 2x + 4 and y =3x-1 and y=4x+1 there is no solution.

There is no pair of (x,y) values that will satisfy all three of these equations simultaneously. Of course, I can _choose_ any two of the equations and get a solution for that pair, and then I could switch to another choice, which would be quite a lot like what happens when a four-legged stool wobbles.

If you had 2 legs of roughly equal length, the stool would rock back and forth, if you held it, and fall over to one side or the other when you let go. Add a third leg the same distance out as the other 2 are apart, and the stool stands by itself, even if the legs are not exactly the same lengths. When you add a fourth leg, you can go back to rocking on two legs, and becoming steady when either of the other two legs hits the floor, preventing it from falling over, but it can rock either way. That's the wobble.

The reason is that a three-legged stool stands on three points, which define a plane, so they are stable, because they are always in a plane. Four points are not always in the same plane.

However, four-legged stools have less chances to tip over.

It appears that chairs and tables, especially wobbling ones are are a nuisance in every household and restaurant. You all are absolutely right. A puzzle point for: Katrina, Tom, Barry G., Dennis, Mommy, annie-marie, SteveGoodman18, kj, Bean (who has identity crisis :), Annie, Donna, Pat, Lynnet, carrie, Andree, TyYann.

And OMG, Wang - thank you for pointing out my mix-up. This was supposed to be Tom's 20th celebration. OK, next week a special puzzle for Tom. Wang - we will count it as your early 30-th puzzle birthday present :)

Now, back to chairs: I need two bar chairs and three counter-top chairs. A builliard store has a surpring sale of fashionable alluminum chairs for $50 each (down from $144). They are four-legged chairs, but the bar height. Shall I buy five of them and cut a few inches off the legs on three to make them counter-top level? I am afraid that it will be almost impossible to cut the legs in such a way that the chairs won't wobble.

Next week - a celebration for kj & anne-marie for their 10 puzzle points and for Tom for his 20 puzzle points.

I've done precisely that (buy bar stools and cut them down to countertop height) myself, and wobbling was no more a problem with the cut-down ones that it had been with the tall ones.

Of course, what you could do would be to go to the hardware store and put adjustable feet (technically, I guess you could get away with a single adjustable foot) on them. They screw up and down to make fine adjustments, and you can generally either attach them to a solid thing or fit into a tube, depending on how the chairs are manufactured.

Ha! No wobbling after chair circumcision! It is reassuring to know that it has been done before and successfully.

I also very much like the idea of one adjustable feet. I had no idea such thing existed. It seems that every manufactured table and 4-legged chair should have one adjustable foot. Lynnet's mom - I think you should patent this.

Thank you for the suggestions!

Hey Maria, the adjustable feet are great IF your floor there is very flat. If it's other than flat, you'll rarely be happy with 4-legged stability. But if the price is right, and you like the color, go for it. Sounds like a bargain. Rubber feet (tips) are also helpful.

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