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令人费解的回形针

Perplexing Paperclips - Numberphile

一张纸 带状的 两个普通的回形针
A strip of paper, a paper ribbon, and two ordinary paper clips.
我们将纸折叠成Z或者S型 像这样
We bend the paper strip in a Z or S shape, like so.
就像这样弯曲
You know, going winding about like that.
然后 在回形针的帮助下将其定型
And, with the help of the paper clips, we shall fix that shape
把回形针固定在这里和这里
by slotting the paper clips here and here.
现在 如果我开始拉纸带会发生什么呢?
Now, what happens if I start pulling the ends of the paper?
是的 你可以看到回形针变得越来越近
Well, you can see that the paper clips come closer and closer together,
而它们是被纸带动的
because they are being driven by the paper.
如果我继续拉 最后它们就会互相接触
So, if I keep pulling, eventually they’ll touch and jam,
如果继续地拉……
but, if I keep pulling nonetheless…
它们跳出来了!
they hop!
当我拿起它们的时候
And, when I pick them up…
它们是连着的 真有趣
they’re linked. That’s quite interesting.
它们连起来了
They’re linked.
我们刚刚做的只是用了两个回形针我们称之为版本零
What we have just done is with two paper clips only, and this what we shall call ‘version number zero’.
这只是一个开始 而这个小把戏可以追溯到很久以前
This is just the beginning. The trick goes back quite a long way;
根据马丁 加纳德的即兴魔法百科全书可以知道
according to Martin Gardner’s Encyclopedia of Impromptu Magic,
它由一位住在华盛顿街的 叫做鲍曼的人发现的
this seems to have been discovered by somebody called Bowman in Washington State.
如今我们在这基础上做了一些小改变
Today we’ll do some variations on this.
依旧 我们把纸弯曲成这个形状
So, we fix the shape of the bending paper like so.
但是 这次我们添加了一个橡皮筋可以看到 纸在这里和这里形成了两个环
But this time we added a rubber band. You see, the paper has two loops here and here,
然后 在你看来 我们把橡皮筋挂在了左边
and, seeing from your side, we’re hanging the rubber band on the left loop.
这是版本一 第一个版本 接着 抓住这里
This is version number 1. Version number 1, then, hanging over here.
会发生什么呢?这次它会这样
What happens? This time what happens is this.
橡皮筋挂在了纸带上
You have the rubber band hanging from the paper ribbon;
而下面的回形针和橡皮筋连在一起
underneath clip-clip together from the rubber band.
我们把这样的叫做不完全连锁——你可以看到为什么称其为不完全连锁
We have this kind of half-chain—and you will see in a moment why we call it ‘half-chain’—
因为橡皮筋和回形针连在一起了
of rubber band-clip-clip together.
科学地说 你特别要锻炼你的想象力和观察力
In science, you have to exercise your power of imagination and observation, especially.
是的 你不能了解一点然后就说:“哦,这没什么”
You know, you can’t just take something and say ‘Oh, this is not going to matter’,
然后忽略它的一点点的改变
and then reject little differences of innocuous appearance.
其实也许你要很注意这一点点改变
Maybe you have to pay a lot of attention to such little differences.
为了说明这一点 我们来做一个很像版本一的
To illustrate that point, let’s do something that looks like version number 1,
像我们刚刚做过的那个 但不是
which we have just tried, but isn’t quite.
在版本一 我们是这样做
Number 1 that we have just tried looked like this
橡皮筋是挂在两个圆环的左边一个
The rubber band was hanging from the left loop of those two loops.
嗯 现在[清嗓子]
Okay, now [clears throat]
橡皮筋是挂在右边的环而不是左边的
The rubber band is now hanging from the right loop instead of left loop,
我们可以叫它为版本二
and that’s what we shall call ‘version number 2’.
你看 它之前是挂在这 现在是挂在这
You see, before it was hanging here, and now it’s hanging here.
这就是版本二
So, that’s version number 2.
如果我一直拉纸带会发生什么?
What’s going to happen if I pull the ends of the paper?
你知道的 不细心的人可能会说:“这有什么区别?”
Well, you know, somebody not observant might say ‘Well, what’s the difference?
是啊 可能没什么区别
Here, here, well, probably doesn’t matter.’
你记得在版本一的最后 橡皮筋挂在了纸带上
But you remember that version number 1 ended up having the rubber band hanging from the paper ribbon,
而两个回形针挂在橡皮筋上
with two paper clips hanging from the rubber band.
这次 在版本二 所有的东西都从纸带上掉下来了
This time—version number 2—whole thing falls off the paper ribbon,
所以 这就是不同 而相同的 就是你拿起它们
so that’s different. Nonetheless, when you pick it up,
得到的还是回形针-回形针-橡皮筋的不完全连锁结构
you still get this clip-clip-rubber band, this half-chain together.
在版本一 我们得到是是这样的
So, in the case of version number 1, we had this,
而在版本二 它们(和纸带)分离开了
and, version number 2, this got detached.
试过了版本一和版本二后
Having tried version number 1 and version number 2,
我们很迫不及待
it becomes tempting, indeed irresistible,
想尝试将版本一和版本一加在一起会怎样
to try both at the same time, what is 1+2.
开心的是我这里有另一条橡皮筋
Well, it’s very lucky that I have another rubber band here.
我猜 你觉得版本一和版本二加在一起应该是这样的
I think, and I’m sure you think too, 1+2 should look like this.
可以看到 这个橡皮筋在本来版本一的位置
You see, because by itself, this rubber band is in the position of version number 1.
被挂在左边
It’s hanging from the left.
而这个 挂在版本二的位置
This one, by itself, is in the position of version number 2,
所以这应该就是版本一加版本二;版本一和版本二会同时进行
so 1+2 should be like this; it’s doing 1 and 2 at the same time.
所以问题是 如果我一直拉纸带会发生什么?
So, the question is what happens if I pull the ends of the paper.
这两条橡皮筋和两个回形针会怎样?
What’s going to happen to the two rubber bands this time, and the two paper clips?
但是在此之前 我先要解释一下
Before we do that, however, let me debunk myself a little bit.
在版本一 橡皮筋挂在了纸带上
The fact that, in version number 1, the rubber band stayed on the paper
而版本二 橡皮筋从纸带上掉下来
and—number 2—the rubber band fell off the paper,
其实这并不神秘 我们可以了解它
is, in fact, no mystery; we can understand this.
所以 我们试着来了解下
So, let’s try to understand it–
舍弃回形针的干扰
by ignoring the paper clips.
还记得吗?版本一是这样的
You remember that version number 1 looked like this?
橡皮筋被挂在了左边的环上
The rubber band was hanging from the paper on the left loop.
如果看仔细点 可以发现是纸带穿过了橡皮筋
If you look carefully, you see that the paper strip goes through the rubber band
然后从一旁出来
and then passes on the side.
也就是说 橡皮筋是和纸带连在一起的
In other words, the rubber band is linked with the paper strip.
所以 如果我一直拉橡皮筋
So, if I pull the ends of the paper…
由于它们是连在一起的 没有办法只能连在一起
of course it stays linked, and it has no choice but to stay on the paper.
这就是为什么在版本一 橡皮筋是留在纸带上的
That is why, in version number 1, the rubber band stayed on the paper.
而版本二是这样的
In contrast, version number 2 looked like this.
橡皮筋被挂在了两个环的左边一个上
Over the two loops in the paper, it was hanging from the right loop.
如果你看仔细 你会发现 纸带从橡皮筋穿过然后又穿了回来
If you look carefully, you see that the paper comes into the rubber band but goes back out.
所以 事实上 橡皮筋并没有和纸带连在一起
So, the rubber band is, in fact, not linked to the paper.
当我一直拉的时候 由于它们没有连在一起所以橡皮筋就掉了下来
When I pull the ends then, being unlinked, the band is free to fall.
这就是为什么版本二中橡皮筋会掉落
That is why the rubber band can fall off the paper in version number 2.
所以 版本一的橡皮筋留在纸上而版本二的掉落
So, the fact that the version number 1 stays on the paper, number 2 falls off the paper,
并不神奇
is no mystery as I say.
我们完全可以分析理解
That’s a purely topological thing that we can understand.
让我们回到版本一加版本二那里
Let’s go back then where we left off and try to do 1+2.
所以版本一和二一起做会发生什么?
So, what’s going to happen if we do 1 and 2 at the same time?
你可以猜猜看
You should pause and try to guess.
Brady:它们全都会连在一起 然后挂在纸上
BRADY: They’re all gonna to be linked, and on the paper.
Tadashi:很好 说得再仔细点——
TADASHI: Good. But, a bit more precisely—
Brady:我猜会形成一条长链
BRADY: I picture one big long chain—
Tadashi:长链 橡皮筋-回形针-回形针-橡皮筋这样的长链?
TADASHI: Long chain. Rubber band-clip-clip-rubber band, in that kind of order?
还是橡皮筋-回形针-橡皮筋-回形针?
Or rubber band-clip-rubber band-clip?
或者可能是两个橡皮筋连在一起?好吧 我只是为了钱
Or, maybe two rubber bands get linked? Well, I do that only for money.
Brady:我不知道 我猜是橡皮筋-回形针-回形针-橡皮筋
BRADY: I don’t know, I picture— my guess is rubber band-paper clip-paper clip-rubber band.
Tadashi:很棒 这很好 猜测是很重要的
TADASHI: Excellent, OK, so that’s quite nice. By the way, it’s always important to guess anyway,
因为猜测是认知的方法 事实上 在科学领域
because, you know, guessing is really the way to learn and, in fact, advance in science,
所有的学者和研究人员都是这样的 当猜对了你会感到特别地自豪
both for students and researchers alike, because if you guess right, you are very, very proud you got it right.
而如果猜错了 你会很惊讶 或者是有点惊讶
And if you guess wrong, you are really shocked; maybe not really, but slightly shocked,
然后你就被吸引住了 就会去了解到底是怎么回事
and that engages your thinking, and you can learn what happened,
而你下次就会更聪明(因为有经验)
and then it makes you a little smarter next time.
所以 在解决问题前要先猜测下
So, always guess before solving any problem.
好 让我们试试这个
OK, let’s try doing this.
我的朋友Brady猜说会是橡皮筋-回形针-回形针-橡皮筋
My friend Brady guessed rubber band-clip-clip-rubber band.
那我们就来试试 版本1+2会发生什么?
Let’s do this. 1+2, what’s going to happen?
开始了
Here we go.
确实没错 恭喜你
Of course that’s what happens, congratulations.
想想其实是有道理的
And it makes sense, in retrospect, because if we didn’t have this rubber band
你看到的是这样的 挂在上面的 当然这是版本一的情况
you see that configuration — the remaining configuration — is, of course, the number 1 configuration.
如果没有这个橡皮筋 你会看到连在一起的回形针-回形针-橡皮筋 是版本二
If we didn’t have this rubber band, well, you get clip-clip-rubber band hanging — that’s number 2 —
而因为没有这个橡皮筋
but that would be because this rubber band is not present,
所以所有的东西就会从纸上脱落下来
would be detached from the paper, so everything will fall down.
而合起来得到的就这个长链
So, together, you get this long chain.
每当我们听到“理论上”时
When we hear about computation, or calculation in general,
总是感到理所当然 因为我们在学校学的就是这样
we always think naturally because that’s how we learn these things in school, and about numbers
——据统计 或者总的来说 理论上等
— calculating with numbers — or, perhaps at a more advanced level — calculating with formulas.
然后 你就做了设计了一些方案
And, you know, you do some algorithm, you do some recipe
接着就等着理论结果出炉
and then the expected results come out.
然而 尽管你的大脑其实还没有正式开工
But here, your brain, although it hasn’t formalised anything,
它没有写任何数字或者公式
you know, it hasn’t written any numbers or formulas,
但却高效的计算了
has effectively computed.
它捕获到了一些非常丰富的模式特征
It has caught on to something- some really rich pattern in nature,
然后就可以开始理解发生了什么
and has started understanding what was going to happen.
你的大脑能够不通过数字或者公式
So, it’s quite curious that your brain is good at computing,
就能够计算 这让人十分好奇
even without numbers or formulas.
好的 如果我在纸带上绑住两个版本一的皮筋
OK, what happens if I take two copies of number 1 on this paper ribbon,
再拉开 结果会是怎样呢
and do the same experiment pulling the ends?
1+1…
1+1…
就像这样
looks like this.
看到没 这条皮筋在原先熟悉的位置1
You see? This rubber band is in the position of now-familiar position number 1.
这条 – 顶部这条 – 可以从背面看到
This rubber band — the top one — as you can see by looking at it from the back,
是版本1的另一种拷贝
is another copy of version number 1.
那么 我们现在有了版本1的两份拷贝 都绑在纸带上
So, we have here two copies of version number 1 living together on this strip of paper.
两条皮筋都连接在纸带上
And you can see that both of those rubber bands are linked to the paper
纸带分别穿过它俩
in the sense that the paper goes through one and through the other.
那么 当我拉到底时 皮筋一定会待在纸上 这点我名可以知道
So, when I pull the ends, the rubber bands must stay on the paper, that much we know.
主要的问题是 它们和回形针会形成什么状态?
The only question is: how do they interact with the paper clips?
愿意猜猜吗?
You care to guess?
BRADY:我觉得两条皮筋都会待在纸带上…
BRADY: I think both rubber bands are gonna stay on the paper…
TADASHI:没错
TADASHI: That’s true.
BRADY:两个回形针会与它们首位连接
BRADY: …and the two paper clips will be linked to each of them on the ends.
TADASHI:两个回形针之间呢?
TADASHI: How about paper clips between themselves?
BRADY:我认为会是 皮筋-回形针-回形针-皮筋
BRADY: I think it’ll be rubber band-paper clip-paper clip-rubber band.
TADASHI:啊 不错哦 那么 我们猜 最终会是一长条 皮筋-回形针-回形针-皮筋
TADASHI: Ah, that’s good. Indeed, we are guessing that we- I can get that long chain of band-clip-clip-band.
我们来试试
Let’s try this.
的确
Indeed.
当然
Of course.
再一次 我们认为自然的行为应该是这样 自然遵守了
Again, we thought nature should behave in a certain way, and nature obliged.
这里你得到了某种悬挂桥
You get this kind of suspension bridge.
橡皮筋挂在纸上 我们对这种拓扑结构早已了解
Rubber bands on the paper, that we had already understood topologically,
但是新鲜的是 这条长链 皮筋-回形针-回形针-皮筋
but what is new is that you get, again, this long chain of band-clip-clip-band
悬挂在纸带上 回形针连接在两条皮筋之间
hanging like this; paper clips connecting the rubber bands in the middle in this hanging configuration.
这就是1+1 那么…
That’s 1+1. So…
1+2 是这样 1是连接在纸上 2不是
1+2 was like this because number 1 is linked to the paper; number 2 isn’t.
1+1的意思就是版本1有两份 都在纸带上
And 1+1 was like this because they are both copies of number 1 and they have to be on the paper.
每次 我们都会得到一条长链 皮筋-回形针-回形针-皮筋
Each time, then, we are getting a long chain- a full chain of band-clip-clip-band,
区别在于 这条长链与纸带是什么连接方式
and the only question is how that long chain is positioned with respect to the paper.
嗯 还剩下 2+2没试过
Well, it remains for us to try 2+2…
版本2 我记得 是会从纸带上掉落的
Version number 2, I’d like to recall, is the one where by itself fell off the paper.
2+2 是这样 因为 你们看 例如
2+2 is like this and that’s because- if you look at, for example, just at this rubber band
这是我们熟悉的版本2
that’s in the now-familiar position of number 2;
纸带穿过去 但是最终会出来
paper goes in but comes back out.
这里 在顶部 有版本2的另一份拷贝
And over here, at the top, you will see another copy of version number 2.
OK 我们现在要做的就是 把它们拉到底来看看会发生什么
OK, so, what we have to do is to pull the ends and see what happens.
额 这一次 两条皮筋都没有再连接到纸袋上
Well, this time, neither of the band, by itself, is linked to the paper.
结果会是怎样?
So, what’s going to happen?
在我拉开之前 尝试这些不同的组合之前
When I was exploring this, and trying various possibilities and some,
在这个阶段 第一次到这个阶段
at this stage, when I came to this stage for the first time.
自然而然的 我猜结果仍然会是一条长链 皮筋-回形针-回形针-皮筋
naturally, I guessed that I would get a long chain of band-clip-clip-band,
和之前一样
which we do.
因为没有一条皮筋连接在纸带上 我想整个长链会掉到地上
And because neither band is linked to the paper, I thought the entire thing will just fall down to the ground,
纸带上什么也不会有
leaving the paper alone.
好 我们来试一下
So, let’s try this.
然而 这次 发生了一些奇怪的事
This time, however, something strange has happened.
这条长链整体挂在了纸带上 没有掉到地上
The entire ensemble of the rubber bands and paper clips didn’t fall down to the ground;
它们绑在了纸带上
instead, they stayed on the paper.
看到没?皮筋用交叉的方式绕在了纸带上
You see? There is a rubber band that goes around the paper like this in a transverse manner.
这里 中间位置 有一根皮筋 扮演了一把锁的角色
And then there is – in the middle – a rubber band, which is kind of acting as a lock, so to speak.
请忽略这里的回形针 本阶段它们没什么意思
Please ignore the paper clips there; not that interesting at this stage.
那么 这个中间的“锁” 这条皮筋交叉着 绕过纸带
So, there’s a middle ‘lock’, and then the band that goes around – transversely – around the paper.
可以看到 如果没有这条皮筋 另一条就会从纸带上被拉下来
You can see that if we didn’t have this rubber band, the other one could be pulled off the paper.
但如果没有另一条
On the other hand, if we didn’t have this rubber band,
这个中间的锁 悬挂在上面的 就会掉落到地上
then this middle lock, which is hanging, would fall down to the ground.
因此 皮筋都没有连接在纸带上 这是事实
So, neither band is linked to the paper, that’s true,
但是它们又都互相阻止对方离开纸带
but each one is preventing the other from leaving the paper.
因此 如果现在我合上纸带的尾部
So that, if I now closed the ends of this paper,
可以认为纸带是第三条环 我们就有了一个三个环的系统
and think of the paper as the third loop, we have here a system of three loops.
其中的任意两个环都不是相互连接的 但这三个
Any two of these are mutually unlinked, but three together…
却绑在了一起
are linked and stuck together.
这种配置有个名字 叫做博罗梅安链
And that kind of configuration has a name: it is called a Borromean link.
博罗梅安链
Borromean link,
用意大利文艺复兴家族 博罗梅奥 来命名
after the Italian Renaissance family called Borromeo—
不不 这不是那个互相下毒的家族 那是 波吉亚家族 是另一个
no, no, it’s not the family who used to poison one another; that’s Borgia, that’s another family—
博罗梅奥
but Borromeo.
这是一个会在各种情况下出现的概念
And it appears in all sorts of contexts.
三维拓扑结构里非常重要的中心物体
Very, very important central object, for example, in 3-dimensional topology.
这是一个博罗梅安链的例子
Here is an example of a Borromean link.
这三块木头的每一块都是一个环
Each of those three pieces of wood is a loop—
尽管 有点变形的环 但的确是环
rather, sort of, distorted loop, but it is a loop nonetheless—
而且它们清晰的拴在一起 你无法把它们分开
and they’re clearly stuck together; you cannot pull them apart.
但是 可以看出来 很神奇 用你内心感受一下
But, you see, it’s curious to note that, if you make in your mind’s eye
这三个中的任何一个消失 例如
any one of these three disappear, by saying ‘kazam’, for example.
我们让这个消失
Let’s make the one that I’m holding disappear.
另外两个就会分离 不再连接
Then the other two can be pulled apart, so they are not linked.
这三个中的任意一个都一样
And it’s the same for any of these three.
因此 如果你把这个拿掉 另外两个就会被拆开
So, if I make this one disappear, the other two can be pulled apart, and
任何一个消失 都会让另外两个散架
any one of these disappearing will cause the other two to fall apart.
这就是所谓的博罗梅安结构
So, that is a Borromean configuration.
这个是从一整块木头上雕刻出来的 当我最初在非洲买到这个时
This is carved out of a single piece of wood, and when I first bought it in Africa,
听到这个说法时 我非常惊讶
I was very impressed to hear this.
但是 被你们发现了 当然这也是刻出这个样式的唯一方法
But then, you caught me, that that’s the only way to carve something like this, of course.
好的 我们已经看到了不少东西了
Good, we have seen quite a few things,
如果你想给亲朋好友们分享 建议你们这么玩
and if you wanted to share this with friends and family, I suggest that you end on the following,
有点邪恶的 请注意
rather mischievous, note.
你先说 好好 你们都明白了 是吧
You say ‘OK, OK, folks, you understand everything, right?’,
这时候 这些人会有些骄傲了 自信了
and by this time, they’re feeling cocky and confident.
会说 是啊是啊 我们都搞懂了然后你说 OK 那我们来回顾一下最开始的那个吧
‘Yeah, yeah, we understand everything’, and you say ‘OK, let’s revise the first thing that we started with’.
看看我把纸带折成Z或者S形
What happens if I bend the paper like this in a Z or S shape,
加上回形针 固定好 然后拉到底 这会怎样?
and, with the help of paper clips, fix them in place, and then pull the ends of the paper?
他们就会说 哦 我们都看过这个啊 会连到一起啊
They say ‘Oh, we have seen all that’, and ‘they get linked together, right?’, and so on.
这很奇怪 因为 不到十分钟之前
It’s really strange because, not more than ten minutes ago,
他们还从未见过回形针能连接在一起呢现在他们却会认为 每一次回形针都会连在一起
they had never seen paper clips come together. Now they assume that, every single time, they come together.
很有趣
That’s quite curious.
那么 你说 好的 但是我们还是要来试一下的
So, you say ‘OK, OK, but let’s try this.’
那么 回形针夹在纸带上 我一拉 啊 奇怪了它们没有连到一起
So, paper clips on paper, and when I pull – ah, that’s strange; they didn’t get linked together.
发生了什么?
What happened?
因为 一些人 也许不包括你 被我刚才的嘟嘟囔囔给分散了注意力
‘Course, some people – maybe not you – were distracted by my patter,
没有观察到
and haven’t been watching.
我们之前做的其实是 把回形针放到Z型纸带的顶部
What we did was, indeed, put the paper clip at the top of this Z shape.
但是另一只 我没有放到顶部 而是放到了另一端的底部
But the other one – instead of putting it at the top, I put it at the bottom on the other side.
这里的回形针
So, paper clip here…
还有这里
and here.
那么 当我拉开的时候 它们当然就分开啦 不可能连到一起
So, when I pull, they’re completely separated; there’s no incentive for these to link together.
因此 你需要仔细观察
So, you have to keep watching.
这只是个小恶作剧 不是所有的都没价值 我们现在基于这个来做实验
But not all is lost. This is just a piece of mischief, but you can start doing science on top of this.
我们试试将回形针放到错误的位置上 但是加上橡皮筋
Let’s try the paper clips in the wrong positions, but with addition of the rubber bands.
我们试试将橡皮筋放到所谓1+1的位置 也就是说
Let’s try rubber bands in the position that we used to call 1+1, in other words:
两条橡皮筋都连接在纸带上
both rubber bands are linked with the paper.
橡皮筋看上去就像是1+1
As far as the rubber bands are concerned, it’s just like 1+1.
既然两条橡皮筋都连在纸带上 它们应该会保持待在纸带上
As you can see both rubber bands are linked to the paper; they have to stay on the paper.
所不同的是 回形针一个在顶部
What’s different from the previous experiment is that one of the paper clips is at the top,
一个在底部
but the other one is at the bottom.
它们应该都在顶部 这样才能神奇的连接到一起
It used to be both at the top, and both at the top – mysteriously, they used to link.
但这次 回形针不会连接在一起
But this time, the paper clips will not be linked.
然而 橡皮筋 我们已经理解拓扑结构
Rubber bands, however – we have already understood the topology, because the paper
因为纸带穿过了橡皮筋 它的会留在纸带上
goes through the rubber band – has to stay on the paper.
那么 拉到底会是什么结果呢?
So, what’s going to happen when I pull the ends of the paper?
额?
Huh?
有人敢猜猜吗?
Any takers? Any guesses?
BRADY:也许会是 每条橡皮筋上挂着一个回形针?
BRADY: Like, maybe, each rubber band will have one paper clip?
TADASHI:哦 有意思
TADASHI: Okay, that’s interesting, so…
我的朋友Brady说 当我拉直纸带时
My friend Brady’s saying, well, when I straighten out the paper,
橡皮筋会悬挂在纸带上 这是必然的
rubber bands will be hanging from the paper; that’s obligatory.
每条橡皮筋都会挂着一个回形针
Each rubber band will have a paper clip, this rubber band, paper clip.
但是回形针不会连接在一起 因为它们不在纸带的同一边
But paper clips will not link because they’re on the wrong sides of the paper.
我们拉开试试
Let’s try this.
漂亮
Excellent.
正是这样 看到没?
That’s what happens. You see?
我们进行了计算
We have started computing.
理解了一些道理 这里其实包含了很多聪明的方法
We understand something. There’s a lot of method to be smart in this.
我想 这很有趣 我们试试
I guess, interesting, let’s try this.
和之前一样
That’s inter— it looks like the previous one
但是这次每个回形针都连在了橡皮筋上 但回形针之间没有连接
but this time, each paper clips is linked to the rubber band, but not between themselves.

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