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你关注的YouTube科普博主都在做这道图论题

Science YouTubers attempting a graph theory puzzle

现在是圣诞节日期间
It’s the holiday season,
是时候把大家聚在一起 做点有意义的事情了
a time of year to bring people together and to do something a little bit different. So…
所以…… Mythologer 我是Matt Parker Standup Math的作者 嘿!
Mathologer here. I’m Matt Parker from Standup Maths.
我是Sam 来自Wendover Productions和Halfinteresting大家好
Hey, this is sam from Wendover Productions and half is
我是Jame Grimes 来自Singing Banana频道
interesting hi everyone this James Grime from the singingbanana channel
Brady集结完毕 我来自Numberphile、Objectivity和好多
Which Brady reporting for service from numberphile objectivity and various other channels.
别的频道 大家好!我叫Steven
Hey everyone, my name is Steven Walsh
Walsh 我的频道是Welsh
My channel is Welch labs.
Labs我来自Looking Glass Universe频道
I’m from the channel Looking Glass Universe.
Grant告诉我他要
Grant told me he, was sending, me a
给我一个题目和一个马克杯
Puzzle and a mug.
嗨Grant!
Hey Grant, I am here I’ve got a mug,
我这里有一个马克杯 一些纸
and some paper and some markers and
一些马克笔 我已经准备好解决这道题目了
I’m ready to do your puzzle I really should know
我绝对应该知道怎样解这个马克杯问题
how to solve this mug, because i’m the guy
因为这东西是我和Matt Parker
that makes and sells them with Matt Parker
一起制作并销售的
so I’ve been instructed not to read the directions before
他们告诉我开始之前不要读题目
Starting I’ve been.
嗨Ben!嗨Grant!
Hey Grant so a friend just gave me
一个朋友刚给我了这个杯子
this mug you are gon na be challenged
你马上要被挑战
And I’m just gon na kind
我让你在摄像机前做这个题目
of make you do this on camera to embarrass you
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
We’ve got three different houses here three different cottages
还有三种不同的资源:煤气、电力和水
and then three different utilities the gas the power and the water draw
将每种资源和每个房屋之间都用线连接起来
A line from each of the three utilities to each
所以一共是九条线
of the three houses so nine lines in total okay without letting
好的 任意两条线都不相交
Any two cross, no two lines crops,
没有任何线相交 嗯
is right here if you, wanted to just
这里如果你想直接从电厂连到
go straight from power to the house
这个房子好的
right Okay,
有意思
interesting that is quite a challenge
这个还挺难的 9条线都不相交
so nine lines that don’t cross that doesn’t even sound possible.
这看上去就不太可能吧 这就是我的马克杯
I’ve got, my mug I’ve got my utilities mug here
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
I’ve even got real coffee in the mug i mean
看看我还挺注重细节的
that look at that that’s attention To,
我还挺想试一试的
detail i’m willing to give this a, go i’m just
我就是不知道
Worried I’m gon na muck it up I
我怎么在这上面画
tend to make bit of a poker square of
我老是……
these things when I when I truck Say,
你看!
well let’s just fill in as
我们先来看看随便画能画几条
many as i can and see what happens i
肯定会
‘m sure this will end terribly.
变得很糟糕
So there’s one?
先画一条
There’s the other
这是另一条接着…
There we go.
…煤气管道 这很简单
Gas line it’s, gonna be easy we’re gonna go like this,
我们这么连 %……&%@¥……%
wow sound effects are crucial,
声音效果是非常重要的
I’m not gon na go
我不要绕过这个绿色的
around the green one don’t want to fall for that
我不想砸在这里 我还能再画一条
I can do another one and now up to five four,
这就是五条了 只要再画四条 我在看我的显示器
go I’m looking at my display over here I should
我应该放在前面就好了 哦
have put it over there but, oh well.
这么画挺好
Oh that’s good of it
这么画……等一下……这么画
That’s your go to the second is okay?
差不离是这样
There’s no ibly this is easy enough
这倒挺简单然后我们只要从这里连到那里
And so we just need to get from here to there.
这是1 2 3 4 5
I have one two three four five six seven lines
67条线
two to go.
还有俩 所以这个连到那个
So I have that one connected to that one I
然后这个连到那个
Mean that one connected to that one.
啊我们现在遇到问题了
Oh now, we get into trouble, okay,
我知道问题是什么了
now I start to see the problem.
就在这里
And there I have made my fatal error
我犯了一个致命错误
in not paying attention I
我把这个房子围起来了
have boxed in this house right here
你看就是这里
as you can see there’s
外面的线连不到
no way to get to it.
里面去了 煤气要连到第二个房子
Gas needs to get to number 1 and 2.
是第一个和第二个 问题就出在这里
And that’s the problem because we’re cut
因为我们要冲破这个
Off i kind of want to try it on paper,
我有点想在纸上试试 好吧
okay it’s getting really
在马克杯上画图有点太奇怪了
Awkward to draw on a mug i think what i’m gon na,
我想我还是应该找
do is i’m gon na go to a
一张纸画一画 杯子具有这样一个性质
Piece of paper this this kind of property that
你可以像这样
you can, make lines go from here to
把线这么连 以及这么转一圈连过来
here and also all the way around
这样说来我们似乎应该在纸上应该画一个
Makes it seem like i should, be drawing a
球面 就像这样
spear Something like that Okay, let me i need, bigger,
好的我们就 我们需要大一点……
lines bigger bigger space
更大的空间 额……唉…… 我这样又把它给绕起来了
But now i’ve just blocked off how is this
这怎么能解得出来呢 这样弄不出来
possible this isn’t getting anywhere let’s try, again
我们再来试试 水要连到第一个和第二个
Water i need, to the first and second
What?!我又给弄乱套啦! 至少让它看着
What i really messed it up okay, to make that at least look,
舒服一点
easier i’m gon na go around
我们这样绕一圈过来
here
到这边绕——绕——绕——绕——到
Around around around around around to to go around
把煤气绕一圈连到这里
the mug with, the gas here, so
那么我来把它
i’m just gon na go all the way
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
Around i’m gonna go around Let’s go underneath the handle here So now it’s closed
现在我们只要想办法
We just need to figure out how,
把红的连过来
to get that red in there
第三号房子 已经一切就位了
house number three is all done and good
看看!
look at that house number three good to
第三号房子搞定了! 那么 这个房子的全部三个
go so this house has all three and
都连上了 那个房子全部三个也都连上了 但是中间的这个
That house has all three but this one in the middle
没有连到煤气 好的吧 我来试试一个新的
doesn’t have gas Alright let, me try something, new
我想在这上面实验一下
Let me just try an experiment here let’s let’s.
让我们搞一点经验主义
Be let’s be empirical
马克杯很好的一点
What’s really nice about the mug
是它非常光滑
Is that it’s shiny so
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
if you use a dry erase marker you can undo your mistakes you rub it off Posit,
暂停!好
okay, so there’s some very
在这个题目里面有些 非常美妙的数学
pleasing math within, this puzzle for you,
我们可以深入聊聊
and me to dive into but first let
开始之前请允许我衷心感谢
Me just say a really big thanks to
所有这些
everyone here, who, was willing to
愿意给我的这次实验当小白鼠的人
be my, guinea pigs in this experiment
他们里面每个人都经营了一个非常受
Each of the runs a channel that i respect
我青睐的频道
A lot and many
他们里面很多人也热情支持本频道那么
of them have been incredibly kind and helpful to this channel
如果你对他们中的任何人不太熟悉的
So if there’s any there that you’re unfamiliar
或者你还没有
with or that you haven’t been keeping
关注到的 他们的信息都在视频描述里
Track with, they’re all listed in the description
你一定一定要关注他们
so most certainly check them out, we’ll get
我们过一会儿在继续看他们的表现
back to all of them in just a minute
让我们来谈一谈这个题目
Here’s the thing, about the puzzle if
如果你要在纸上尝试的话
you try it on a piece of paper you
那就会遇到麻烦了
‘re gon na have a, bad time
但如果你有一个数学家的头脑 当一道题看上去
But if you’re a mathematician at heart when a puzzle seems hard.
很难的时候
You don’t just throw.
就不会甩手而去
Up your hands and walk, away
而会去尝试 解决一个所谓
Instead you try to solve a meta puzzle of
“超越问题”
sorts see if you can, prove that the
比如看看你能不能证明你遇到的问题是
task in front of you is impossible
不可能解出的 在这个例子中
In this case how on earth do you,
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?
do that how, do you prove something is impossible
一点背景知识:
For background anytime that you have
当你遇到一些东西和这些东西
Some objects with a notion
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
of connection between those objects it’s called a graph often represented abstractly with dots for your objects
我们称之为”顶点“ 和代表其连线的线段
Which i’ll call vertices and lines for your connections,
我们称之为”边
which i’ll call edges
“ 在多数情况下你怎么画这个图都不
Now in most applications the way you draw
影响什么 而
A graph, doesn’t matter what matters
只有他们的连线是重要的 但在一些特定的问题上
is the connections but in some peculiar cases
就比如本题
Like this one the thing that we care about
我们关注的是如何画这个图
is how it’s drawn and if you can
如果你能在一个平面内把图画出来
draw a graph in the plane without crossing
并保证其各条边不相交 它就叫作”平面图
Its edges it’s called a planar graph
“ 所以我们要解的这个题就是
So the question before us is
我们这个资源连线问题
whether or not our utilities puzzle graph
也就是有逼格的数学家们称作 ”
Which in the lingo is fancifully called a
完全二分图K33“是不是一个平面图此刻
complete bipartite graph k33 is planar or not
我们
And at this point there are two kinds
有两种观众
of viewers those of you who know
一种是懂欧拉公式的 另一种是不懂的
About euler’s formula and those, who don’t those, who?
懂欧拉公式的观众大概明白我要说啥
Do might see where this is going
不过我不打算直接
but rather than pulling out a formula
给出公式 然后用这个公式解决”超越问题
from thin air and using it to solve the meta puzzle i
“ 我打算倒过来 来说明一步步
Want to flip things around here and show. How
解决这道难题的过程中
Reasoning through, this conundrum step,
能够让你重新发现一个非常 有魅力还很普适的数学结论
by step can lead you to rediscovering a very charming and very general piece of math
我们开始 你在这些房子
To start as you’re drawing
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
Lines here between homes and utilities one really important thing to keep note
当你围起来一个新的区域的时候
of is whenever you enclose a new region
也就是这个油漆桶工具每次涂上
that is some area that the paint bucket tool,
颜色的部分
would fill in
因为你看 你每次围上这样
Because you see once you’ve enclosed a region,
一个区域的时候 新的线就没法
like that, no new, line that you draw
从外面连进来或从里面连出去
Will be able to enter or exit it
所以对这些区域要慎重
so you have to be careful with these
还记不记得上个视频里面
In the last video remember how.
我提到过 一个解决问题的技巧
I mentioned that a useful problem-solving tactic is to shift
就是把关注的焦点
Your focus onto,
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?
any new constructs that you introduce trying to reframe your problem around them
在这次的问题里
Well in this case, what can, we
这些区域能提供什么思路?
say about these regions right now i have up
现在屏幕上的是一个未完成的解法
on the screen and in complete puzzle
自来水还没有连
Where the water is not yet connected to
到第一个房子 这里有四个不同的区域
the first house and it has four separate regions
但是你能不能预测 假如问题得解
But can, you say anything about how. Many regions
那 这个解的图里面有多少个区域?
A hypothetically complete puzzle would have what about the number
每个区域由多少条边围成?
of edges that each region touches, what can you say there
你能推测吗? 这里面你可能会发现很多东西
There’s lots of questions you might, ask
也可以提出很多问题
And lots of things you might notice and
如果你幸运的话 可能会想到这个问题:
if you’re lucky here’s one thing that might pop out for a new,
对于你画的每条能围出区域的线
line that you draw to create a
它一定会连到一个
region it has to hit a vertex that already
连过其他边的顶点上
has an edge coming out of it
来 我们这么想
Here think of it like this start by
开始想象 其中一个顶点是”点亮“的
imagining one of your nodes as lit up,
而其它五个
while the other five are dim and
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
then every time you draw an edge from a lit up vertex to a dim vertex light up the, new,
的顶点点亮了
one
所以一开始 每条新的边点亮一个新顶点
So at first each new, edge lights up one more vertex
但是 如果你连到了一个
But if you connect to an
已经点亮的顶点 你会发现这么做
already lit up vertex notice how
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:
This closes off a new region and this gives us a super useful fact,
每条新的边
each new, edge either
要么就将点亮的顶点的数量加一
increases the number of lit up nodes by one
要么就将包围的区域的数量加一
or it increases the number of enclosed regions, by one
通过这个事实
This fact, is something that,
我们就可以算出 一个假设存在的
we can, use to figure out the number of regions that a?
解会将整个平面分成多少个区域
Hypothetical solution to this would cut,
你知道怎么做么?
the plane into can, you see how
开始的时候在这个空间中你有
When you start off there’s one node
一个已经亮着的顶点
lit up and one beaten all of duty’ space
和一个新被点亮的顶点
By the end we’re going to need, to draw.
你需要画九条线
Nine lines since each
因为这三种资源 各自要和三个房子相连
of the three utilities gets connected to each of the three houses
其中五条线
Five of those lines are going
会用来 点亮那些开始的时候
to light up the initially dim vertices
暗着的顶点 所以剩下的四条线 每条都要产生一个新的区域所以
So the other four lines, each must introduce a new region
一个假定存在的解
So a hypothetical solution would cut.
会把平面分成五个不同的区域
The plane into, five separate regions and you might say, okay,
你可能会说:”
that’s a
好的
Cute fact but,
那是一个不错的事实 可问题怎么就无解了呢?
why should that make things impossible what’s wrong with having five regions
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
Well again take a look at this partially complete graph notice
请注意:每个区域都由四条边所围成
that each region, is bounded by four edges
实际上对这个图来说 你不可能
And in fact for this graph you could never have a cycle with,
画出一个不足
fewer than four edges
四条边的区域 假设你从一个房子开始
Say you start at a house then the next line
下一条线就得连到一种资源上
has to be to some utility and then a line
然后从那里连出来的线就要连到另一个房子
out of that is going to go to another house and You,
在然后你不能直接连会你最开始连出去的那里因为
can’t cycle back to where you started immediately
你在连回最开始那个房子之前必须
because you have to go to another utility before you can
先连到另一个资源上
Get back to that first house
所以每个圈都必须
So all cycles have at least four edges and this
至少有四条边 就是这里使我们能够证明我们的原题无解
right here gives us enough to prove the impossibility of our puzzle Having,
要分出五个区域
five regions, each with a boundary of
每个区域都需要有至少四条边 会需要超过我们可以
at least four edges would require more edges than, we have available
画的边数 我们现在来画
Here let me draw.
一个平面图 和我们的资源问题
A planar graph that’s completely different
完全不同但可以帮助我们理解五个区域
from our utilities puzzle but useful for illustrating what, five regions with
每个区域有四条边意味着什么
Four edges each, would imply if you went
如果你遍历每个区域
through each of these regions, and add up
并把其各自边的数量相加
the number of edges that it has
那么你就会
Well you end up
得到五乘以四 也就是二十
with five times four or twenty and of course this
当然这么数的话 就把图中所有的边数
Way over counts the total number of edges
给数多了 因为 每条边都围着不止
in the graph since each edge is touching multiple regions
一个区域 实际上每条边被且仅被两个区域共享
But in fact each edge is touching exactly two regions
所以20这个数字 就恰好是边数的二倍
so this number twenty is precisely double counting the edges So,
所以任何把平面分成五个区域
any graph that cuts, the plane into,
且每个区域包含四条边的图
five regions, where each region is touching four
就一定要有总共十条边
edges would have to have ten total edges
然而 我们的资源问题只能连起来九条边
But our utilities puzzle has only nine edges available
所以尽管我们得出了我们必须把平面
So even though, we concluded that it would have
分成五个区域才能解出
to cut, the plane into, five regions it
我们却根本不可能分出
would be impossible for her to do that
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
So there you go bada-boom bada-bing it is impossible to solve this puzzle
这个题的 除非让线相交
on a piece of paper without intersecting lines tell
这可不是投机取巧的解法
me that’s not a slick proof, and
在我们回到我们的朋友们和马克杯们之前
Before getting back to our friends and the
我们不妨花点时间从中
mug it’s worth taking a moment to pull out
抽提出一个普适的结论 回想我们的关键结论
A general truth sitting inside of this think back to the key rule,
即每个新的
where each, new
边要么连到一个没有被连过的点上
Edge was introducing either a new vertex
从而引入一个新的顶点 要么它引入一个新的区域
by being drawn to an untouched spot or it introduced a new enclosed region
同样对逻辑对任何平面图都成立
That same logic applies to any planar graph,
并不只限于我们的资源谜题的情况
not just our specific utilities puzzle situation
换句话说 顶点的数量
In other words the number of vertices minus the number
减去边的数量 加上区域的数量 总是不变的
of edges plus the number of regions remains unchanged No,
无论你画的图是什么样
matter what graph you draw,
具体地 这个值从2开始 便保持为2
namely it started at two so it always stays at 2 in this relation
这个对任何平面图都适用的关系
True for any planar graph is called euler’s
被称为“欧拉示性方程”
characteristic formula Historically,
在历史上 这个方程提出的对象是凸多面体
by the way the formula came up in the context
比如立方体这样的
of convex polyhedra, like a cube for example
这里顶点数减边数加
Where the number of vertices minus the number
“面”数
of edges plus the number of faces always equals two
总是等于2
So when you see it written down.
所以它写出来的时候往往用F
You often see it with an f
来表示“面” 而不说“区域”
for faces instead of talking about regions
不要以为我和小学寒假作业
Now before you go thinking of me as some
的编者一样 给朋友们出一些无解
kind of grinch that sends friends an impossible puzzle
的题目还让他们拍摄解题过程
and then makes them film themselves trying to,
请注意
solve it keep in mind i didn’t, give,
我没有把这个问题 写在纸上寄给他们的哟!
this puzzle to people on a piece of paper
我相信这和杯子把手有什么关系
And i’m betting the handle has something to do with this. Ok,
不然的话 你干嘛还拿一个
otherwise, why, would you have brought a, bug over here
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
This is a valid observation Maybe use the mug handle, oh? Yeah,
用马克杯的把手 哦我大概知道了
i think i see okay i feel
我感觉可能跟把手有点关系
like it has to do something with the handle
这样我们可以让一条线跳起来越过另一条
And that’s our ability to hop one line
我大概要开始来
over the other i’m gon na start by i think
利用这个把手
Taking advantage of the handle because i think
因为我感觉这应该是解决问题的关键
that that is the key to this you know
怎么说呢
what i think actually a sphere
我有点感觉球面可能是一个错误的思路因为
is the wrong thing to be thinking about i
那么有个著名的结论 马克杯在拓扑上和甜甜圈等价
Mean like famously a mug is topologically the same as a
那么要解决这一问题
Doughnut so to solve this thing you’re
你需要利用
Gon na have to use the torus enos
马克杯的“环面性质” 你需要用到把手
of the mug you can have to use the handle somehow
从而构造出环面
That’s the thing that makes
我们用绿色
this a torus mm-hmm let’s take the green
越过这里的这个把手
and go Over the handle here okay?
然后红色就可以这么从下面过去
And then the red can kind of come under nice
-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
My approach is to get as far as you can
能连多少
with
连多少
As far as you can as
就像在平面
if you are on a plane
上解决这个问题
and then See,
然后看看卡在了什么地方 看
where you get stuck so look i’m gon na draw
我还要把 这个连到这里
this too, here like that and Now i’ve come
现在我们遇到了一个问题
across a problem because electricity
因为电力连不到这个房子
Can’t be joined to this house this
这就是你需要用到把手
is where you have to use the handle so whatever you
的地方了
Did do it again but go around the handle,
现在需要把你做过的再重做一遍
so i’m gon na go down here
我要从这边下来 我要从下面
I’m gon na loop Underneath come back around,
穿过它转回来
and back to where i started
回到开始的地方 那么现在 我就有办法
And now i’m free to get my electricity
把我的电力 连连连
messy there you,
连到这儿然后
go and then i’m gon na go on the inside of
我要跑到把手的背面
the handle go all the way
绕一大圈到把手背面
around the inside of the handle and
最终连到
finally connect To,
煤气公司
the gas company to solve this puzzle
要解决这一问题 先画一个M型
just drawing the m. And there’s three more
好 还有三条线要连
connections to go so let’s just make them
就这么画
one
12……我还要把
Two and i will have to connect those,
这俩连起来看
two guys right just watch it
从前门进去 后门出来
In through the front door out. Through the back, door done No,
完成!
intersections
没有交叉大概你会说这样是作弊好吧
Maybe you think that it’s cheating, well
这是个拓扑学问题 所以
sort of topological puzzles so it means the relative
图中的相对位置不影响结论
positions of things, don’t matter what that
这就意味着 我们可以把把手拿掉
Means is we can, take this handle and move it here
挪到这里这就出现了另一个连线
Creating another connection, oh? Oh,
嚯嚯嚯 哦天呐 我做出来了吗?
my, god am i done is
就这样完成了?
this over i think i might’ve gotten 24
我貌似用了24分钟 Grant告诉我只要
minutes granny says to take 15 minutes
15分钟的 快看!我想我已经解出来啦!
There you go i think i’ve solved it you haven’t success but but,
-你还没做出来呢?-是很难 但并非无解
not impossible hard but not impossible this
是不是这样的?大概这可能
Isn’t it maybe perhaps not the most elegant
不是这问题最好看的解法
solution to this problem and if i drew this
我在这里画一条线 你会说:
line here you’ll think, oh? No,
“不! 这会把房子圈起来
he’s blocked that house there’s No,
煤气就过不去了!”
way to get the gas
这就是我们为什么要用
in but this is why it’s not a mug right because if you take The,
马克杯了 因为 你只要把煤气线这么连到顶上去
gas line all the way up here to the top.
然后
You then take it over and
翻过来进到马克杯里面
into the mug if you draw
如果你把线画到 咖啡下面——这会把笔弄湿
The line under the coffee it wets the pen
然后等笔再出来的时候
so when the line comes back out, again,
它就画不出线了
the pens not working anymore you can Go,
你可以直接穿过这条线把它连上去
straight across there in and join it up and
因为笔画不出来线了
because it wasn’t drawing you haven’t.
所以你也没有让线交叉
Had across the lines Baby,
简单! 等下
by the way funny story so
这场面真够搞笑 最开始有人送我这个杯子
i was originally given, this mug as
当作礼物 我并不知道它
a gift and i didn’t really know
是哪里买的 直到我开始邀请
Where it came from and it was only after
人们一起制作这个视频之后
i had invited people to be a part of
我才意识到这个
this that i realized the origin of?
杯子是来自 Math Gear网站
The mug maths kheer is a website run By,
这网站是我邀请的三位YouTube作者
three of the youtubers i had
Matt、James和Steve运营的
just invited matt james and steve small world given just how.
世界真小!
Helpful these
你看他们供应的这些东西为我
Three guys, were and the logistics of a
帮了大忙
lot of this really the least i could,
我能感谢他们的方式就是
do to thank them, is give a
展示给你
Small plug for how,
MathGear的礼品卡 可以作为一个挺不错的还来得及买的圣诞礼物
gift cards from matt’s gear could, make a pretty good last-minute christmas present
回到问题 这种问题就是
Back to the puzzle though this is one of
那种一旦你知道答案
those things where once you see it it kind
就会感觉非常显然的问题
of feels obvious the handle of the
杯子的把手可以当作一个桥梁
Mug can, basically be used as a
来避免两条线相交
bridge to prevent two lines from crossing,
那么这就带来了一个非常
but this raises a really interesting mathematical question
有趣的数学问题 我们刚刚证明了这个问题
We just proved that this task is impossible
在平面上是无解的
for graphs on a plane so where exactly
那么这个定理对于马克杯的表面来说
does that proof break down on the surface
具体在哪个环节上出了问题?
of a mug and
不过我不会在这里告诉你答案
I’m actually not going to tell you the
我希望你自己
answer here i want you to think about this
来想一想
on your own and i don’t just mean
我的意思不是:
saying
“哦!就是因为欧拉公式在有洞的曲面
Oh it’s because euler’s formula is different
上不一样!” 真的
on surfaces with the whole really think about this
请想一想我刚刚展示的逻辑链条究竟的
Where specifically does the line of reasoning
哪一具体环节
that i laid out break down
在马克杯的情况下被破坏了?
When you’re working on a mug i promise you
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
thinking this through will give you a deeper understanding of math Like,
就像任何一个解决难题的人一样
anyone tackling a tricky problem you will likely run
你非常可能会碰壁
into walls and moments of frustration
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
But the smartest people i know actively seek out new,
即使这调整仅仅是益智游戏
challenges even if they’re just toy puzzles They,
他们提出新的问题
ask, new questions they aren’t afraid to start
他们勇于一次次地重新开始 他们拥抱每个失败的时刻
over many times and they embrace every moment of failure So,
所以请用心尝试这个
give this and other puzzles and
和其它的难题 永远不要停止
earnest try and never stop, asking questions
提出问题 “那么 Grant”我
But grant i hear you complaining how,
听到你抱怨 “我怎么练习解题技巧呢
am i supposed to practice my problem-solving if i don’t have
又没有人给我” “寄来在拓扑奇异的
Someone shipping me puzzles on topologically interesting shapes,
表面上的题目”最后
well let’s close things off by, going,
我们来看几个由这周
through a, couple puzzles created By,
的数学相关赞助商 brilliant.org
this week’s mathematically oriented sponsor brilliant dork
提供的题目
So here i’m
所以这是他们的解题技巧入门课程
in there intro to problem solving course and going Through,
这个系列的问题叫“翻硬币” 其规则大概是
a particular sequence called flipping pairs and the rules here seem to be that we can, flip,
我们可以同时翻转
adjacent
相邻的两个硬币 但是不能一个
Pairs of coins, but, we can’t flip, them one at
一个地翻转 要回答的问题是
A time, and we are asked is it possible to get it so
有没有可能 翻成三个硬币同时金色
that all three coins are gold side up
一面朝上的情况 显然我刚刚翻出来这个情况了
Well clearly i just did it so yes
所以 Yes 下一个问题我们从一个不同的排布模式开始
And the next question, we start with different configuration,
规则是一样的
have the same rules and rask the same question can
问题也是一样的
we get it so that all three
能不能翻出三个金色面的情况然后
of the coins are gold side up and
你看我们这里没有多少自由度
You know there’s not really that many degrees of freedom,
只能点两个不同的地方
we have here just two different spots to click so you
你可能会很快得到结论:不
Might quickly come to the conclusion that no you can’t even if you,
办不到你可能还不知道理论上
don’t necessarily know the theoretical reason
的原因为何 这也没关系
Yet that’s totally fine so,
选No
no and we kind of move along?
你就可以继续
So next it’s kind
然后下面它给出了全部可能的 初始排布 并问
of showing us every possible starting configuration that there is and asking for how
这其中有多少
Many of them can, we get it to a point,
我们能 翻出三个金色面朝上的情况
where all three gold coins are up

Obviously i’m kind of giving
我有点泄露答案了 就是右面这个4
Away the answer it’s sitting here four on the right
因为我已经做过这题了
because i’ve gone through this before but if you
不过如果你想自己做一做的话 这个问题
Want to go through it yourself this particular quiz
的解决过程很有趣
has a really nice resolution and a lot of
而且这个课程的
others in this course do build up
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
Genuinely good problem-solving instincts so you can, go to brilliant org Slash,
.org/3b1b 他们就可以知道你
three b1 b2 let, them know that you came
从这个视频过来 或者访问/3b1b
From here or even slash 3 b 1
/flipping 直接跳到这个问题 建立账户是免费的
b flipping to jump straight into this quiz and you can
他们的很多内容都是免费的
Make an account for free a lot
但是如果你想获得
of what they offer is free but They,
全部体验的话 他们也有年费
also have a annual subscription service if you
用户选项
want to get the full suite of
我觉得他们很不错
Experiences that they offer and i just think they’re
我认识那里的一些人
really good i know a couple of the people there and they’re
他们在 如何组织数学解释方面也是十分用心的
Incredibly thoughtful, about how. They put together math explanations
自来水连到1
water goes to one and then wraps
然后包过来
around to the other and
到这边在这个地方 我naive地 哦 等下
Naively at this point, oh, wait i’ve already messed up
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!
Then from there water can, make its way to cut it number three.
我被困住了! 我又做错了!
Ah i’m trapped i’ve done this wrong, again

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https://www.youtube.com/watch?v=VvCytJvd4H0

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