beautiful weather, and you have ordered the beer,

the beer comes, you put it in a table, and then you touch the table

and the table is unstable and the beer is poured out.

You are angry!

It’s a four-legged table. The table is completely stable.

The problem is the ground on which the table stands.

This is not flat and that’s why one leg is above the ground.

And then if you put your hands again on the table,

it goes down and it’s the instability of the table.

The moment solution is you take a sheet of paper.

For example this paper is under the beer glass and put it under this leg

and for a while, it looks okay but after a few minutes, we are angry again because of this paper

is compressed a little bit and instability again.

And we hate that.

Mathematicians never have unstable tables.

They know what to do.

And what you do is very very simple.

Turn the table and start moving the table and try to turn it

so that you have a quarter of a turn and on the way of your turning,

there will be a moment where it’s absolutely stable.

So you’re just rotating like a rotator like rotating a disc?

Yes, I rotate the table like a disc and typically only a few centimeters are needed

and suddenly it’s stable and this is not by chance.

This there it’s a mathematical proof that this will always happen.

You’re gonna have to give me that proof now.

I give you that proof now.

Here’s the ground and here this is the position of the four legs

and we enumerate them… this is leg 1, this is 2

this is 3, this is leg 4.

And suppose that leg 1 is above the floor whereas these three are fixed on the ground.

Now, of course, if we put pressure on 1, then we still the instability.

And now, we do the following:

We measure the height of leg 1.

Remember, we always measure the height of leg 1.

So if you do that in time, then we get associate it to time T,

we associate height of leg 1.

So time is zero, we get some T=0,

we get some number say X>0.

Now nothing is happening, now let’s start moving

and we do that obviously in time and at each time, we measure the height of leg 1.

And we turn it in this way.

All we turn it so that we try to bring leg 1 to the position of leg 2.

At each time, we measure the height of leg 1.

So this gives the function f (t).

For each time, T we measure.

Here’s something important.

It can happen that if we fix It all 2, 3 and 4,

this is all you remember, we fix 2, 3 and 4 and now it could happen that the height of leg 1 is negative.

Yeah because this will happen, if we now put leg 1 into position of leg 2,

leg 2 to the position of leg 3, leg 3 to the position of leg 4

and leg 4 is at the position of leg 1.

But now, we remember that we fix the position of 2, 3 and 4.

I fixed them on the ground.

I keep them on the ground.

And since we did it here, at this position, this was above the ground,

and now we force these three to be on the ground.

That means this position has to be under the ground.

You see that before you fix these three, now you force these to go down,

and this is suddenly under the ground.

So it’s time 1, let’s suppose take time into 1 until at this position,

so it’s t=1, this height is negative.

So now, we draw if I can get you a sheet of paper.

Now we draw this curve so this is time = 0, this is time=1

Here we draw the height and at times zero, the height was something positive.

And at time =1, the height was negative.

So this is f (0), this is f (1).

And now, at each time T, we get the position of the height and you see we get a curve

and it might go even up and down but in the very end, it has to end here.

And now comes the famous theorem of Mathematics, the Intermediate Value Theorem

which just says that if you have a continuous function which is positive here and negative here,

they match the opposition here where it is 0.

It could be multiple ~It could be… you can have fun with it, you turn your table further

and it might be in the 2nd position of this table.

You don’t need that but it’s fun to try that out.

And if you are in the beer garden,

and if you do it the next time in the beer garden, you will easily fix the table.

And you will be pleased and you can taste it even better.

I’ll do it all the time whenever I’m in the beer garden or even in a restaurant,

often the ground is not flat, and I sit there with my friends

and they are saying, “Ahh, let’s put this under.”

I said, “Don’t do it!”

I move it just a little bit and they are obviously very surprised.

And we do not change and it’s fix for the whole evening.

What if the tables are all lined-up or with special shape or something?

Oh, that’s of course… Mathematics is always theoretical so if you cannot move the table,