beautiful weather, and you have ordered the beer,

天氣非常好，然後你點了一杯啤酒，

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

啤酒來了，你把它放在桌上，可是你一碰桌子，

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

桌子不穩，啤酒就打翻了。

You are angry!

你超級生氣！

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

這是一張四腳桌，桌子非常平穩，

The problem is the ground on which the table stands.

問題卻出在供桌子站立的地面。

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

地面不平，所以有一隻桌腳總是懸在半空中。

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

如果再把你的手放到桌上，

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

它又靠到地上了，所以它就是桌子不平的最大原因。

The moment solution is you take a sheet of paper.

現在的解決辦法就是拿一張紙。

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

舉例來說，拿啤酒杯墊來墊桌腳，

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

剛開始看起來很有用，但在幾分鐘後，我們會因為這張紙

is compressed a little bit and instability again.

被壓扁了，讓桌子再次不平穩而生氣。

And we hate that.

我們非常討厭那樣。

Mathematicians never have unstable tables.

數學家絕不會有不平穩的桌子。

They know what to do.

他們知道該怎麼做。

And what you do is very very simple.

你要做的事情其實非常簡單。

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

轉動桌子，移動桌子，然後試著轉動它，

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

那麼你就會有一個四分之一的圓圈，而且當你在轉動的時候，

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

一定會找到一個地方是絕對平穩的。

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

所以你就像一個旋轉軸一樣轉動，像轉動唱片那樣嗎？

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

對，我就像轉唱片一樣轉動桌子，我只需要幾公分的距離，

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

一下子它就平穩了，而且這不是偶然發生。

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

這有數學證明的，這常常會發生。

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

你現在證明給我看。

I give you that proof now.

我現在就證明給你看。

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

這是地面，然後這個是四隻腳的位置，

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

我們把它們點出來，這是第一隻，這是第二隻，

this is 3, this is leg 4.

第三隻，第四隻。

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

假設第一隻腳懸在半空中，其他三隻卻都靠在地上。

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

當然，如果我們用手壓在第一隻腳上，桌子還是不會平。

And now, we do the following:

所以現在，我們要這樣做：

We measure the height of leg 1.

測量第一隻腳的長度。

Remember, we always measure the height of leg 1.

一定要記住我們是要量第一隻腳的長度。

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

加入時間因子，把它命名為T，

we associate height of leg 1.

與腳1的長度作關聯。

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

所以時間為零，我們就寫T=0。

we get some number say X>0.

我們得到的數字就寫X>0。

Now nothing is happening, now let’s start moving

現在什麼事情都還沒發生，開始轉動桌腳吧！

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

我們一定要馬上做，而且每一次，我們都要量第一隻腳的長度。

And we turn it in this way.

我們現在轉向這裡。

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

我們要轉桌腳，所以我們要把腳1轉到腳2的位置。

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

每個時間我們都要測量腳1的長度。

So this gives the function f (t).

所以這時候就有功能f(t)了。

For each time, T we measure.

每一次我們都要算時間。

Here’s something important.

這很重要。

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

如果我們校正腳2、腳3、腳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.

請記住，我們校正腳2、腳3、腳4可能會發現，腳1的長度成為負值。

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

因為接下來我們會把腳1換到腳2的位置，

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

腳2換到腳3，腳3換到腳4，

and leg 4 is at the position of leg 1.

腳4換到腳1的位置。

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

但現在，我們要記得已經校正過腳2、腳3和腳4的位置。

I fixed them on the ground.

它們的長度都已經被修正到剛好可以靠在地上。

I keep them on the ground.

它們都剛好靠在地上。

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

因為我們把它換到這裡，這個位置，所以它懸在半空中，

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

現在，我們要把這三個腳都靠在地上。

That means this position has to be under the ground.

那表示這個位置必須要在地平面下。

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

在你校正這三支腳的長度前，他們被迫往下，

and this is suddenly under the ground.

突然，這隻腳就會埋入地底。

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

所以現在是第一個時間，假設到這個位置才是第一個時間，

so it’s t=1, this height is negative.

所以時間是1，這個長度不理想。

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

現在，我來找張紙畫給你看。

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

我們現在來畫曲線，這裡時間為0，這裡時間為1

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

我們把長度畫下來了，時間為0，長度可能是理想的。

And at time =1, the height was negative.

時間為1，長度不理想。

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

這是函數0，這是函數1

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

每個時間，我們都能達到桌腳長度的位置，而且我們能到了一個曲線，

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

曲線可能會在高一點或低一點，但在最後一定會在這裡結束。

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

而有名的數學定理─中間值定理就出現了，

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

這表示如果函數有連續功能，這裡是理想，這裡是不理想的，

they match the opposition here where it is 0.

那他們就能在時間為0時達到正確的位置。

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

它可以是多樣化的，它可以，你會覺得它很有趣，你再繼續轉桌子，

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

它可能會是這張桌子的第2個位置。

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

你不需要第2個位置，但值得一試。

And if you are in the beer garden,

而且如果你在露天啤酒店裡，

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

下次你到露天啤酒電石這樣試試，你會更容易修好桌腳高低不一致的問題。

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

你就會非常開心，也會腳的啤酒更好喝了。

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

每次我到露天啤酒店或是餐廳，我都會轉動桌子來測量。

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

通常地面都不平，所以我和我朋友坐在那裡，

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

他們都會說：「把這個店在桌腳下吧！」

I said, “Don’t do it!”

我都會說：「不行！」

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

我稍微移動桌子，他們就會非常驚訝。

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

其實我們並沒有做什麼事，一個晚上就修好了。

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

那如果桌子排成一長列，或是其他特別的形狀呢？

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

當然可以，數學是一種理論，所以如果你不能移桌子，