
Chinese: 
现在是圣诞节日期间，是时候把大家聚在一起
做点有意义的事情了，所以……
Mythologer
我是Matt Parker，Standup Math的作者
嘿！我是Sam，来自Wendover Productions和Halfinteresting
大家好，我是Jame Grimes，来自Singing Banana频道
Brady集结完毕，
我来自Numberphile、Objectivity和好多别的频道
大家好！我叫Steven Walsh，我的频道是Welsh Labs
我来自Looking Glass Universe频道
Grant告诉我他要给我一个题目和一个马克杯
嗨Grant！我这里有一个马克杯，一些纸，一些马克笔
我已经准备好解决这道题目了
我绝对应该知道怎样解这个马克杯问题
因为这东西是我和Matt Parker一起制作并销售的
他们告诉我开始之前不要读题目
嗨Ben！嗨Grant！
一个朋友刚给我了这个杯子

English: 
It's the holiday season, a time of year to bring people together and to do something a little bit different so
Mathologer, here I'm Matt Parker from stand up maths. Hey, this
Is sam from Wendover Productions and half is interesting hi everyone this James Grime from the singingbanana channel
Which Brady reporting for service from numberphile objectivity and various other channels. Hey everyone, my name is Steven Walsh
My channel is Welch labs. I'm from the channel Looking Glass Universe. Grant told me he, was sending, me a
Puzzle and a mug. Hey Grant, I am here I've got a mug, and some paper and some markers and
I'm ready to do your puzzle I really should know
how to solve this mug, because i'm the guy
that makes and sells them with Matt Parker so I've been instructed not to read the directions before
Starting I've been. Hey Grant so a friend just gave me this mug you are gonna be challenged
And I'm just gonna kind of make you do this on camera to embarrass you

Chinese: 
你马上要被挑战，我让你在摄像机前做这个题目就是为了调戏你。我们这里有三个不同的房子，就是三个小木屋
还有三种不同的资源：煤气、电力和水
将每种资源和每个房屋之间都用线连接起来
所以一共是九条线
好的
任意两条线都不相交
没有任何线相交
嗯
这里，如果你想直接从电厂连到
这个房子
好的，有意思，这个还挺难的
9条线都不相交，这看上去就不太可能吧
这就是我的马克杯
我的市政管道杯
好的，这就是我自制的马克杯
我还真的在杯子里倒上了咖啡，看看我还挺注重细节的
我还挺想试一试的
我就是不知道我怎么在这上面画，我老是……
你看！
我们先来看看随便画能画几条，肯定会
变得很糟糕。先画一条
这是另一条
接着……煤气管道，这很简单

English: 
We've got three different houses here three different cottages and then three different utilities the gas the power and the water draw
A line from each of the three utilities to each of the three houses so nine lines in total okay without letting
Any two cross, no two lines crops, is right here if you, wanted to just go straight from power to the house
right
Okay, interesting that is quite a challenge so nine lines that don't cross that doesn't even sound
possible. I've got, my mug I've got my utilities mug here
I've even got real coffee in the mug i mean that look at that that's attention
To, detail i'm willing to give this a, go i'm just
Worried I'm gonna muck it up I tend to make bit of a poker square of these things when I when I truck
Say, well let's just fill in as many as i can and see what happens i'm sure this will end terribly. So there's one?
There's the other
There we go. Gas line it's, gonna be easy we're gonna go like this,

English: 
wow sound effects are crucial,
I'm not gonna go around the green one don't want to fall for that
I can do another one and now up to five four, go I'm looking at my display over here I should
have put it over there but, oh well. Oh that's good of it
That's your go to the second is okay?
There's no ibly this is easy enough
And so we just need to get from here to there. I have one two three four five six seven lines
two to go. So I have that one connected to that one I
Mean that one connected to that one. Oh now, we get into trouble, okay,
now I start to see the problem. And there I have made my fatal error in not paying attention I
have boxed in this house right here as you can see there's
no way to get to it. Gas needs to get to number 1 and 2. And that's the problem because we're cut
Off i kind of want to try it on paper, okay it's getting really

Chinese: 
我们这么连
%……&%@￥……%
声音效果是非常重要的
我不要绕过这个绿色的，我不想砸在这里
我还能再画一条，这就是五条了，只要再画四条
我在看我的显示器，我应该放在前面就好了
哦，这么画挺好
这么画……等一下……这么画
差不离是这样
这倒挺简单
然后我们只要从这里连到那里
这是1，2，3，4，5，6，7条线
还有俩。所以这个连到那个
然后这个连到那个
啊我们现在遇到问题了，我知道问题是什么了
就在这里，我犯了一个致命错误
我把这个房子围起来了，你看就是这里
外面的线连不到里面去了。
煤气要连到第二个房子，是第一个和第二个。问题就出在这里
因为我们要冲破这个。我有点想在纸上试试

Chinese: 
好吧，在马克杯上画图有点太奇怪了
我想我还是应该找一张纸画一画
杯子具有这样一个性质，你可以像这样
把线这么连，以及这么转一圈连过来
这样说来我们似乎应该在纸上应该画一个
球面
就像这样
好的，我们就
我们需要大一点……更大的空间
额……唉……
我这样又把它给绕起来了，这怎么能解得出来呢
这样弄不出来
我们再来试试
水要连到第一个和第二个
What？！我又给弄乱套啦！
至少让它看着舒服一点
我们这样绕一圈过来
到这边
绕——绕——绕——绕——到
把煤气绕一圈连到这里
那么我来把它这样绕一圈……绕过来……
从这边的把手下面绕过去
转——转——转——转——
现在好像差不多了
现在我们只要想办法把红的连过来

English: 
Awkward to draw on a mug i think what i'm gonna, do is i'm gonna go to a
Piece of paper this this kind of property that you can, make lines go from here to here and also all the way around
Makes it seem like i should, be drawing a
spear
Something like that
Okay, let me i need, bigger, lines bigger bigger space
But now i've just blocked off how is this possible this isn't getting anywhere let's try, again
Water i need, to the first and second
What i really messed it up okay, to make that at least look, easier i'm gonna go around
here
Around around around around around to to go around the mug with, the gas here, so i'm just gonna go all the way
Around i'm gonna go around
Let's go underneath the handle here
So now it's closed
We just need to figure out how, to get that red in there

English: 
house number three is all done and good look at that house number three good to go so this house has all three and
That house has all three but this one in the middle
doesn't have gas
Alright let, me try something, new
Let me just try an experiment here let's let's. Be let's be empirical
What's really nice about the mug
Is that it's shiny so if you use a dry erase marker you can undo your mistakes you rub it off
Posit, okay, so there's some very pleasing math within, this puzzle for you, and me to dive into but first let
Me just say a really big thanks to everyone here, who, was willing to be my, guinea pigs in this experiment
Each of the runs a channel that i respect
A lot and many of them have been incredibly kind and helpful to this channel
So if there's any there that you're unfamiliar with or that you haven't been keeping
Track with, they're all listed in the description so most certainly check them out, we'll get back to all of them in just a minute

Chinese: 
第三号房子，已经一切就位了，看看！
第三号房子，搞定了！
那么，这个房子的全部三个都连上了，那个房子全部三个也都连上了
但是中间的这个
没有连到煤气
好的吧，我来试试一个新的
我想在这上面实验一下
让我们搞一点经验主义
马克杯很好的一点是它非常光滑
所以你要是用那种白板笔的话
你可以直接把错误的线抹掉
暂停！好，在这个题目里面有些
非常美妙的数学，我们可以深入聊聊。
开始之前请允许我衷心感谢所有这些
愿意给我的这次实验当小白鼠的人
他们里面每个人都经营了一个非常受我青睐的频道
他们里面很多人也热情支持本频道
那么，如果你对他们中的任何人不太熟悉的
或者你还没有关注到的
他们的信息都在视频描述里，你一定一定要关注他们
我们过一会儿在继续看他们的表现
让我们来谈一谈这个题目

Chinese: 
如果你要在纸上尝试的话，那就会遇到麻烦了
但如果你有一个数学家的头脑
当一道题看上去很难的时候
就不会甩手而去，而会去尝试
解决一个所谓“超越问题”，比如
看看你能不能证明你遇到的问题是不可能解出的
在这个例子中，你到底应该怎么做？到底应该怎样
才能证明这件事情不可能？
一点背景知识：当你遇到
一些东西和这些东西之间的连线的时候
这就叫作“图”。通常抽象地表示成
代表那些东西的点，我们称之为”顶点“
和代表其连线的线段，我们称之为”边“
在多数情况下你怎么画这个图都不影响什么，而
只有他们的连线是重要的。但在一些特定的问题上
就比如本题，我们关注的是如何画这个图
如果你能在一个平面内把图画出来，并保证其各条边不相交
它就叫作”平面图“
所以我们要解的这个题就是，我们这个资源连线问题
也就是有逼格的数学家们称作
”完全二分图K33“是不是一个平面图

English: 
Here's the thing, about the puzzle if you try it on a piece of paper you're gonna have a, bad time
But if you're a mathematician at heart when a puzzle seems hard. You don't just throw. Up your hands and walk, away
Instead you try to solve a meta puzzle of sorts see if you can, prove that the task in front of you is impossible
In this case how on earth do you, do that how, do you prove something is impossible
For background anytime that you have
Some objects with a notion of connection between those objects it's called a graph often represented abstractly with dots for your objects
Which i'll call vertices and lines for your connections, which i'll call edges
Now in most applications the way you draw
A graph, doesn't matter what matters is the connections but in some peculiar cases
Like this one the thing that we care about is how it's drawn and if you can draw a graph in the plane without crossing
Its edges it's called a planar graph
So the question before us is whether or not our utilities puzzle graph
Which in the lingo is fancifully called a complete bipartite graph k33 is planar or not

Chinese: 
此刻，我们有两种观众
一种是懂欧拉公式的，另一种是不懂的
懂欧拉公式的观众大概明白我要说啥
不过我不打算直接给出公式
然后用这个公式解决”超越问题“，我打算倒过来
来说明一步步解决这道难题的过程中
能够让你重新发现一个非常
有魅力还很普适的数学结论
我们开始。你在这些房子和资源之间画线的过程中
你应该注意到的一个非常重要的事情是
当你围起来一个新的区域的时候
也就是这个油漆桶工具每次涂上颜色的部分
因为你看，你每次围上这样一个区域的时候
新的线就没法从外面连进来或从里面连出去
所以对这些区域要慎重
还记不记得上个视频里面，我提到过
一个解决问题的技巧，就是把关注的焦点
放在你新构造出来的结构上去
并以此为对象重新分析问题？

English: 
And at this point there are two kinds of viewers those of you who know
About euler's formula and those, who don't those, who?
Do might see where this is going
but rather than pulling out a formula from thin air and using it to solve the meta puzzle i
Want to flip things around here and show. How
Reasoning through, this conundrum step, by step can lead you to rediscovering a very charming and very general piece of math
To start as you're drawing
Lines here between homes and utilities one really important thing to keep note of is whenever you enclose a new region
that is some area that the paint bucket tool, would fill in
Because you see once you've enclosed a region, like that, no new, line that you draw
Will be able to enter or exit it so you have to be careful with these
In the last video remember how. I mentioned that a useful problem-solving tactic is to shift
Your focus onto, any new constructs that you introduce trying to reframe your problem around them

English: 
Well in this case, what can, we say about these regions right now i have up on the screen and in complete puzzle
Where the water is not yet connected to the first house and it has four separate regions
But can, you say anything about how. Many regions
A hypothetically complete puzzle would have what about the number of edges that each region touches, what can you say there
There's lots of questions you might, ask
And lots of things you might notice and if you're lucky here's one thing that might pop out for a
new, line that you draw to create a region it has to hit a vertex that already has an edge coming out of it
Here think of it like this start by imagining one of your nodes as lit up, while the other five are dim and
then every time you draw an edge from a lit up vertex to a dim vertex light up the, new, one
So at first each new, edge lights up one more vertex

Chinese: 
在这次的问题里，这些区域能提供什么思路？
现在屏幕上的是一个未完成的解法
自来水还没有连到第一个房子
这里有四个不同的区域
但是你能不能预测，假如问题得解，那
这个解的图里面有多少个区域？
每个区域由多少条边围成？你能推测吗？
这里面你可能会发现很多东西，也可以提出很多问题
如果你幸运的话，可能会想到这个问题：
对于你画的每条能围出区域的线
它一定会连到一个连过其他边的顶点上
来，我们这么想。开始想象
其中一个顶点是”点亮“的，而其它五个
是”暗的“
然后你每次从一个点亮的顶点
连一条边到一个暗的顶点，你就把这个暗的顶点点亮了
所以一开始，每条新的边点亮一个新顶点

English: 
But if you connect to an already lit up vertex notice how
This closes off a new region and this gives us a super useful fact, each new, edge either
increases the number of lit up nodes by one
or it increases the number of enclosed regions, by one
This fact, is something that, we can, use to figure out the number of regions that a?
Hypothetical solution to this would cut, the plane into can, you see how
When you start off there's one node lit up and one beaten all of duty' space
By the end we're going to need, to draw. Nine lines since each of the three utilities gets connected to each of the three houses
Five of those lines are going to light up the initially dim vertices

Chinese: 
但是，如果你连到了一个已经点亮的顶点，
你会发现这么做就会围出来一个新的区域
这会告诉我们一个超级有用的事实：
每条新的边要么就将点亮的顶点的数量加一
要么就将包围的区域的数量加一
通过这个事实，我们就可以算出
一个假设存在的解会将整个平面分成多少个区域
你知道怎么做么？
开始的时候在这个空间中你有一个已经亮着的顶点
和一个新被点亮的顶点
你需要画九条线，因为这三种资源
各自要和三个房子相连
其中五条线，会用来
点亮那些开始的时候暗着的顶点

English: 
So the other four lines, each must introduce a new region
So a hypothetical solution would cut. The plane into, five separate regions and you might say, okay, that's a
Cute fact but, why should that make things impossible what's wrong with having five regions
Well again take a look at this partially complete graph notice that each region, is bounded by four edges
And in fact for this graph you could never have a cycle with, fewer than four edges
Say you start at a house then the next line
has to be to some utility and then a line out of that is going to go to another house and
You, can't cycle back to where you started immediately because you have to go to another utility before you can
Get back to that first house
So all cycles have at least four edges and this right here gives us enough to prove the impossibility of our puzzle

Chinese: 
所以剩下的四条线
每条都要产生一个新的区域
所以，一个假定存在的解
会把平面分成五个不同的区域。你可能会说：”好的，那是一个不错的事实，
可问题怎么就无解了呢？有五个区域又怎样嘛！“
于是我们再看一下这个还没有解出来的图
请注意：每个区域都由四条边所围成
实际上对这个图来说，你不可能
画出一个不足四条边的区域
假设你从一个房子开始，下一条线就得连到一种资源上
然后从那里连出来的线就要连到另一个房子
在然后你不能直接连会你最开始连出去的那里，因为
你在连回最开始那个房子之前必须
先连到另一个资源上。所以每个圈都必须
至少有四条边
就是这里使我们能够证明我们的原题无解

Chinese: 
要分出五个区域，每个区域
都需要有至少四条边
会需要超过我们可以画的边数
我们现在来画一个平面图，和我们的资源问题
完全不同但可以帮助我们理解五个区域
每个区域有四条边意味着什么
如果你遍历每个区域，并把其
各自边的数量相加
那么你就会得到五乘以四
也就是二十。当然这么数的话
就把图中所有的边数给数多了，因为
每条边都围着不止一个区域
实际上每条边被且仅被两个区域共享
所以20这个数字，就恰好是边数的二倍
所以任何把平面分成五个区域
且每个区域包含四条边的图
就一定要有总共十条边
然而，我们的资源问题只能连起来九条边
所以尽管我们得出了我们必须把平面

English: 
Having, five regions, each with a boundary of at least four edges would require more edges than, we have available
Here let me draw. A planar graph that's completely different from our utilities puzzle but useful for illustrating what, five regions with
Four edges each, would imply if you went through each of these regions, and add up the number of edges that it has
Well you end up with five times four or twenty and of course this
Way over counts the total number of edges in the graph since each edge is touching multiple regions
But in fact each edge is touching exactly two regions so this number twenty is precisely double counting the edges
So, any graph that cuts, the plane into, five regions, where each region is touching four edges would have to have ten total edges
But our utilities puzzle has only nine edges available

Chinese: 
分成五个区域才能解出，我们却根本不可能分出
五个区域。所以，吧嗒噗吧嗒崩，就搞定了
结论是在一张纸面上是解不出这个题的
除非让线相交。这可不是投机取巧的解法。
在我们回到我们的朋友们和马克杯们之前
我们不妨花点时间从中抽提出一个普适的结论。回想我们的关键结论
即每个新的边要么连到一个没有被连过的点上
从而引入一个新的顶点
要么它引入一个新的区域
同样对逻辑对任何平面图都成立
并不只限于我们的资源谜题的情况
换句话说，顶点的数量
减去边的数量，加上区域的数量
总是不变的，无论你画的图是什么样
具体地，这个值从2开始，便保持为2
这个对任何平面图都适用的关系
被称为“欧拉示性方程”
在历史上，这个方程提出的对象是凸多面体
比如立方体这样的

English: 
So even though, we concluded that it would have to cut, the plane into, five regions it would be impossible for her to do that
So there you go bada-boom bada-bing it is impossible to solve this puzzle on a piece of paper without intersecting lines tell
me that's not a slick proof, and
Before getting back to our friends and the mug it's worth taking a moment to pull out
A general truth sitting inside of this think back to the key rule, where each, new
Edge was introducing either a new vertex by being drawn to an untouched spot or it introduced a new enclosed region
That same logic applies to any planar graph, not just our specific utilities puzzle situation
In other words the number of vertices minus the number of edges plus the number of regions remains unchanged
No, matter what graph you draw, namely it started at two so it always stays at 2 in this relation
True for any planar graph is called euler's
characteristic formula
Historically, by the way the formula came up in the context of convex polyhedra, like a cube for example

Chinese: 
这里顶点数减边数加“面”数
总是等于2
所以它写出来的时候往往用F来表示“面”
而不说“区域”
不要以为我和小学寒假作业的编者一样
给朋友们出一些无解的题目还让他们拍摄解题过程
请注意，我没有把这个问题
写在纸上寄给他们的哟！
我相信这和杯子把手有什么关系
不然的话
你干嘛还拿一个马克杯过来
干嘛不拿张纸
这是个有意义的发现
我有个好主意，貌似
用马克杯的把手。哦我大概知道了
我感觉可能跟把手有点关系
这样我们可以让一条线跳起来越过另一条
我大概要开始来
利用这个把手
因为我感觉这应该是解决问题的关键
怎么说呢
我有点感觉球面可能是一个错误的思路

English: 
Where the number of vertices minus the number of edges plus the number of faces always equals two
So when you see it written down. You often see it with an f for faces instead of talking about regions
Now before you go thinking of me as some kind of grinch that sends friends an impossible puzzle and then makes them film themselves trying
to, solve it keep in mind i didn't, give, this puzzle to people on a piece of paper
And i'm betting the handle has something to do with this. Ok, otherwise, why, would you have brought a, bug over here
This is a valid observation
Maybe use the mug handle, oh?
Yeah, i think i see okay i feel like it has to do something with the handle
And that's our ability to hop one line over the other i'm gonna start by i think
Taking advantage of the handle because i think that that is the key to this you know
what i think actually a sphere is the wrong thing to be thinking about i

Chinese: 
因为，那么有个著名的结论
马克杯在拓扑上和甜甜圈等价
那么要解决这一问题，你需要利用
马克杯的“环面性质”
你需要用到把手从而构造出环面
我们用绿色
越过这里的这个把手
然后红色就可以这么从下面过去
-搞定了！-搞定咯！
-我想我做出来了！-没错！
哇噢
我的办法就是尽量多画
能连多少连多少
就像在平面上解决这个问题
然后看看卡在了什么地方
看，我还要把
这个连到这里
现在我们遇到了一个问题
因为电力连不到这个房子

English: 
Mean like famously a mug is topologically the same as a
Doughnut so to solve this thing you're
Gonna have to use the "torus-ness" of the mug you can have to use the handle somehow
That's the thing that makes this a torus mm-hmm let's take the green
and go
Over the handle here okay?
And then the red can kind of come under nice
My approach is to get as far as you can
with
As far as you can as if you are on a plane
and then
See, where you get stuck so look i'm gonna draw
this
too, here like that and
Now i've come across a problem because electricity
Can't be joined to this house this is where you have to use the handle so whatever you

Chinese: 
这就是你需要用到把手的地方了
现在需要把你做过的再重做一遍
我要从这边下来
我要从下面
穿过它
转回来，回到开始的地方
那么现在，我就有办法
把我的电力
连连连
连到这儿
然后，我要跑到
把手的背面
绕一大圈到把手背面
最终连到
煤气公司
要解决这一问题，先画一个M型。好，还有三条线要连
就这么画
1
2……我还要把这俩连起来，看
从前门进去，后门出来，完成！
没有交叉
大概你会说这样是作弊
好吧，这是个拓扑学问题，所以
图中的相对位置不影响结论

English: 
Did do it again but go around the handle, so i'm gonna go down here
I'm gonna loop
Underneath come back around, and back to where i started
And now i'm free to get my electricity
messy there you, go and then i'm gonna go on the inside of
the handle go all the way around the inside of the handle and
finally connect
To, the gas company to solve this puzzle just drawing the m. And there's three more connections to go so let's just make them
one
Two and i will have to connect those, two guys right just watch it
In through the front door out. Through the back, door done
No, intersections
Maybe you think that it's cheating, well sort of topological puzzles so it means the relative positions of things, don't matter what that

Chinese: 
这就意味着，我们可以把把手拿掉，挪到这里
这就出现了另一个连线
嚯嚯嚯
哦天呐，我做出来了吗？
就这样完成了？
我貌似用了
24分钟
Grant告诉我只要15分钟的
快看！我想我已经解出来啦！
-你还没做出来呢？-是很难，但并非无解
是不是这样的？大概这可能
不是这问题最好看的解法
我在这里画一条线，你会说：“不！
这会把房子圈起来，煤气就过不去了！”
这就是我们为什么要用马克杯了，因为
你只要把煤气线这么连到顶上去，然后
翻过来进到马克杯里面。如果你把线画到
咖啡下面——这会把笔弄湿，然后等笔再出来的时候
它就画不出线了
你可以直接穿过这条线把它连上去
因为笔画不出来线了，所以你也没有让线交叉。简单！

English: 
Means is we can, take this handle and move it here
Creating another connection, oh?
Oh, my, god am i done
is this over i
think i might've gotten
24 minutes granny says to take 15 minutes
There you go i think i've solved it you haven't success but but, not impossible hard but not impossible this
Isn't it maybe perhaps not the most elegant solution to this problem and if i drew this line here you'll think, oh?
No, he's blocked that house there's
No, way to get the gas in but this is why it's not a mug right because if you take
The, gas line all the way up here to the top. You then take it over and into the mug if you draw
The line under the coffee it wets the pen so when the line comes back out, again, the pens not working anymore you can
Go, straight across there in and join it up and because it wasn't drawing you haven't. Had across the lines

English: 
Baby, by the way funny story so i was originally given, this mug as a gift and i didn't really know
Where it came from and it was only after i had invited people to be a part of this that i realized the origin of?
The mug maths kheer is a website run
By, three of the youtubers i had just invited matt james and steve small world given just how. Helpful these
Three guys, were and the logistics of a lot of this really the least i could, do to thank them, is give a
Small plug for how, gift cards from matt's gear could, make a pretty good last-minute christmas present
Back to the puzzle though this is one of those things where once you see it it kind of feels obvious the handle of the
Mug can, basically be used as a bridge to prevent two lines from crossing, but this raises a really interesting mathematical question
We just proved that this task is impossible for graphs on a plane so where exactly
does that proof break down on the surface of a mug and
I'm actually not going to tell you the answer here i want you to think about this on your own and i don't just mean
saying
Oh it's because euler's formula is different on surfaces with the whole really think about this

Chinese: 
等下，这场面真够搞笑。最开始有人送我这个杯子
当作礼物，我并不知道它是哪里买的
直到我开始邀请人们一起制作这个视频之后
我才意识到这个杯子是来自
Math Gear网站，这网站是我邀请的三位YouTube作者
Matt、James和Steve运营的
世界真小！
你看他们供应的这些东西为我帮了大忙
我能感谢他们的方式就是展示给你
Math Gear的礼品卡
可以作为一个挺不错的还来得及买的圣诞礼物
回到问题。这种问题就是那种一旦你知道答案
就会感觉非常显然的问题
杯子的把手可以当作一个桥梁
来避免两条线相交
那么这就带来了一个非常有趣的数学问题
我们刚刚证明了这个问题在平面上是无解的
那么这个定理对于马克杯的表面来说
具体在哪个环节上出了问题？
不过我不会在这里告诉你答案
我希望你自己来想一想
我的意思不是：“哦！就是因为欧拉公式在有洞的曲面上不一样！”
真的，请想一想

English: 
Where specifically does the line of reasoning that i laid out break down
When you're working on a mug i promise you thinking this through will give you a deeper understanding of math
Like, anyone tackling a tricky problem you will likely run into walls and moments of frustration
But the smartest people i know actively seek out new, challenges even if they're just toy puzzles
They, ask, new questions they aren't afraid to start over many times and they embrace every moment of failure
So, give this and other puzzles and earnest try and never stop, asking questions
But grant i hear you complaining how, am i supposed to practice my problem-solving if i don't have
Someone shipping me puzzles on topologically interesting shapes, well let's close things off by, going, through a, couple puzzles created
By, this week's mathematically oriented sponsor brilliant dork
So here i'm in there intro to problem solving course and going
Through, a particular sequence called flipping pairs and the rules here seem to be that we can, flip, adjacent
Pairs of coins, but, we can't flip, them one at

Chinese: 
我刚刚展示的逻辑链条究竟的哪一具体环节
在马克杯的情况下被破坏了？
我保证，认真思考这个问题会让你对数学有一个更深的理解
就像任何一个解决难题的人一样
你非常可能会碰壁，遭遇暂时的失败
然而我认识的最聪明的人总是主动寻求挑战
即使这调整仅仅是益智游戏
他们提出新的问题
他们勇于一次次地重新开始，他们拥抱每个失败的时刻
所以请用心尝试这个，和其它的难题
永远不要停止提出问题
“那么，Grant”我听到你抱怨，
“我怎么练习解题技巧呢，又没有人给我”
“寄来在拓扑奇异的表面上的题目”
最后，我们来看几个由这周的数学相关赞助商
brilliant.org提供的题目
所以这是他们的解题技巧入门课程
这个系列的问题叫“翻硬币”
其规则大概是，我们可以同时翻转相邻的两个硬币
但是不能一个一个地翻转

English: 
A time, and we are asked is it possible to get it so that all three coins are gold side up
Well clearly i just did it so yes
And the next question, we start with different configuration, have the same rules and rask the same question can
we get it so that all three of the coins are gold side up and
You know there's not really that many degrees of freedom, we have here just two different spots to click so you
Might quickly come to the conclusion that no you can't even if you, don't necessarily know the theoretical reason
Yet that's totally fine so, no and we kind of move along?
So next it's kind of showing us every possible starting configuration that there is and asking for how
Many of them can, we get it to a point, where all three gold coins are up
Obviously i'm kind of giving
Away the answer it's sitting here four on the right because i've gone through this before but if you
Want to go through it yourself this particular quiz has a really nice resolution and a lot of others in this course do build up
Genuinely good problem-solving instincts so you can, go to brilliant org/3b1b
to them know that you came

Chinese: 
要回答的问题是，有没有可能
翻成三个硬币同时金色一面朝上的情况
显然我刚刚翻出来这个情况了，所以，Yes
下一个问题我们从一个不同的排布模式开始
规则是一样的，问题也是一样的
能不能翻出三个金色面的情况
然后，你看我们这里没有多少自由度
只能点两个不同的地方
你可能会很快得到结论：不，办不到
你可能还不知道理论上的原因为何，这也没关系
选No，你就可以继续
然后下面它给出了全部可能的
初始排布，并问
这其中有多少，我们能
翻出三个金色面朝上的情况
嗯，我有点泄露答案了
就是右面这个4，因为我已经做过这题了
不过如果你想自己做一做的话，这个问题
的解决过程很有趣，而且这个课程的
其他问题会帮助你建立非常好的解题直觉
所以你可以访问brilliant.org/3b1b
他们就可以知道你从这个视频过来

Chinese: 
或者访问/3b1b/flipping，直接跳到这个问题
建立账户是免费的，他们的很多内容都是免费的
但是如果你想获得全部体验的话
他们也有年费用户选项
我觉得他们很不错
我认识那里的一些人，他们在
如何组织数学解释方面也是十分用心的
自来水连到1
然后包过来，到这边
在这个地方，我naive地
哦，等下，我已经弄乱了，嘻嘻嘻嘻嘻
然后从这里
自来水可以穿到第三个小房子那里
啊！我被困住了！
我又做错了！

English: 
From here or even slash 3 b 1 b flipping to jump straight into this quiz and you can
Make an account for free a lot of what they offer is free but
They, also have a annual subscription service if you
want to get the full suite of
Experiences that they offer and i just think they're really good i know a couple of the people there and they're
Incredibly thoughtful, about how. They put together math explanations
water goes to one and then wraps around to the other and
Naively at this point, oh, wait i've already messed up
Then from there water can, make its way to cut it number three. Ah i'm trapped i've done this wrong, again
