4 Dogs Problem: The Inefficient Solution (3b1b SoME#2 Entry)

Summary

在这段视频中，作者分享了一个有趣的数学谜题：四只狗从正方形的顶点出发，以恒定速度相互追逐，最终在中心相遇。尽管作者花了几周时间才找到自己的解法，但发现网上的解答更简洁。通过实验和图表，作者解释了狗的螺旋运动路径，并证明了它们所走的曲线总长与原正方形的边长相等。视频强调了解决问题过程的重要性，并鼓励观众用不同的方法解决一个问题，而不是单一方法解决多个问题。

Takeaways

• 🧩 一道花费了很长时间才解决的有趣谜题，但最终发现更简单的解决方案。
• 🕵️‍♂️ 四只狗分别位于边长为一单位的正方形的顶点，并以每秒一单位的速度分别朝向下一只狗移动。
• 🔄 狗的路径不是圆形，而是向中心慢慢螺旋状移动。
• 🧠 假设狗的反应速度有滞后时间 t 秒，以观察它们的路径变化。
• 📉 建立图表，将每一步的距离变化和中间正方形的边长绘制在坐标轴上。
• 🔬 通过图表可以发现路径长度和中间正方形边长之间的关系。
• 📏 即使没有螺旋方程，仍可以通过图表找到狗移动的总距离。
• 📈 证明在 t 接近零时，图表曲线是直线，且斜率为常数。
• 🧮 计算得出总路径长度等于初始正方形的边长。
• ✨ 强调解决问题的过程和思维方法的重要性，比单纯找到答案更有意义。

Q & A

视频开头提到的谜题是什么？

视频开头提到的谜题是关于四只狗坐在边长为1个单位的正方形顶点上，且每只狗都以1单位/秒的速度朝向下一只狗移动的问题。

谜题中四只狗的移动速度是多少？

四只狗的移动速度是1单位/秒。

四只狗的移动方向是怎样的？

第一只狗总是朝向第二只狗移动，第二只狗总是朝向第三只狗移动，第三只狗总是朝向第四只狗移动，第四只狗总是朝向第一只狗移动。

四只狗最终会在什么地方相遇？

四只狗最终会在正方形的中心相遇。

四只狗的路径会是什么形状？

四只狗的路径是螺旋状逐渐向中心靠拢。

如何通过实验模拟狗的路径？

通过设定狗有一个t秒的延迟时间，每隔t秒调整一次方向，可以模拟出螺旋状的路径。随着t值减小，路径逐渐逼近实际情况。

如何测量狗的螺旋路径的长度？

通过建立一个图表，在x轴上标记每一步的距离，在y轴上标记每一步形成的中间正方形的边长，可以找到狗走到中心的总距离。

如何证明原始情况下的路径是一条直线？

通过计算延迟情况下的路径斜率，发现其斜率趋于负一，证明了原始路径是一条直线。

图表中y轴为0时对应的x轴值是什么意义？

y轴为0时对应的x轴值是狗走到中心的总距离，在本例中等于1单位。

视频的核心主题是什么？

视频的核心主题是解谜的过程本身，而非结果。通过不同方法解决一个问题比单一方法解决多个问题更有价值。

Outlines

00:00

😀 解决数学谜题的个人旅程

作者分享了一个自己花了很长时间才解决的谜题的经历，并描述了在找到更短的解决方案后感到沮丧。然而，作者意识到最短的解决方案并不总是最美的，强调了欣赏解决问题过程的重要性。接着，作者提出了一个关于四只狗沿着正方形顶点移动的谜题，并鼓励观众暂停视频自行尝试解决。

05:00

🤔 狗的螺旋路径和距离计算

作者详细描述了狗在正方形顶点移动的路径，并通过一个思维实验来理解它们的运动方式。假设狗的方向每隔t秒才会调整一次，作者演示了随着t值减小到零，狗的路径逐渐变成螺旋形。然后，作者提出了计算曲线距离的问题，并指出通过图形来解决这个问题的方法。

10:02

🌀 通过图表理解狗的运动路径

作者继续使用思维实验，通过绘制图表来分析狗的运动路径。每一步在x轴上标记一个t距离，在y轴上标记对应的中间正方形的边长。通过这种方法，作者展示了如何利用图表来计算狗移动的总距离，并最终发现，当t趋近于零时，图表实际上是一条直线。这表明狗移动的总距离与正方形的边长相等。

Keywords

💡数学难题

数学难题是指具有挑战性且需要复杂思考和分析才能解决的问题。视频开头提到作者花了数周时间才解决了一个数学难题，显示出这种问题的复杂性和需要的耐心。

💡狗的运动路径

视频中的四只狗在正方形的顶点上，它们每只狗的移动方向总是朝向下一只狗。这个概念展示了一种复杂的运动路径分析，最终它们会在某个点相遇。这种路径的分析是视频的核心问题之一。

💡螺旋

螺旋是一种曲线，在视频中用于描述狗的运动路径。尽管这些狗不会沿着圆形路径移动，但它们会沿着类似螺旋的路径逐渐靠近中心。

💡时间间隔

时间间隔指狗在每个固定的时间段后改变方向。视频中用这个概念来简化和分析狗的运动路径，从而更好地理解它们的最终路径。

💡图形

视频中提到通过绘制图形来帮助理解狗的运动路径。这个图形显示在每一步中狗所形成的中间正方形的边长，并将其与步数关联。

💡极限

极限在数学中是指一个变量逐渐逼近某个值。在视频中，通过减少时间间隔，分析了当时间间隔趋近于零时狗的运动路径，得出它们的路径实际上是一条直线。

💡直线

尽管狗的运动路径看起来复杂，但通过分析发现，当时间间隔趋近于零时，它们的路径实际上形成了一条直线。这是视频中一个重要的数学结论。

💡毕达哥拉斯定理

毕达哥拉斯定理用于计算直角三角形的边长。在视频中，用于解释两次狗形成的正方形之间的关系，帮助理解狗的运动路径。

💡函数

视频中提到绘制图形时不需要知道螺旋路径的函数，这意味着即使在没有明确方程的情况下，仍然可以通过分析运动路径来得到有用的结论。

💡数学美

作者强调即使解决问题的路径较长，过程中的探索和发现仍然充满了美感。视频传达了数学不仅仅是求解问题，更重要的是欣赏解决问题的过程。

Highlights

接下来，我将为你们分享一个非常有趣的谜题，这是一个让我花了很长时间才解决的难题。

我在网上找到了一个非常简短的解决方案，这让我感觉自己很愚蠢，因为我没有想到更简单的方法。

我意识到，最短的解决方案不一定是最美的。我们走的路可能有点长，但只要我们欣赏旅程，它本身就是美丽的。

现在想象你有四只狗，它们坐在边长为一单位的正方形的顶点上。这些狗的速度总是每秒一单位。

第一只狗总是朝着第二只狗移动，第二只狗总是朝着第三只狗移动，第三只狗朝着第四只狗移动，第四只狗又朝着第一只狗移动。

所有狗同时开始移动，它们应该在某个时刻相遇。问题是它们的路径是什么样的，一只狗为了到达相遇点走了多远。

狗不会沿圆周运动，它们会慢慢螺旋向中心移动。

狗的大脑很慢，它们只能在每t秒后改变方向。在每一步之后，它们形成一个比前一步小的中间正方形。

原始的狗应该没有任何滞后时间，它们应该立即朝着正确的方向移动。

狗在x轴上移动t单位的距离，在y轴上移动中间正方形的边长。

这个图表非常有用，知道任何一步的边长，我们可以找到狗到达那一步所走的距离。

中心处的中间正方形的边长为零，这意味着答案是图表在x轴上为零的点。

对于滞后情况的图表，斜率是负一，这是一个常数。

知道斜率和第一个正方形的点，我们可以轻松找到直线方程。

狗走的曲线长度与正方形的边长相等，结果是惊人的美丽。

解决一个问题的五种方法要比解决五个问题的一种方法更好。

Transcripts

in a moment i'll be sharing with you a

very nice puzzle one which i spent my

head on for a long time in fact it took

me weeks to solve it but i felt really

good about myself that i finally did it

but then i decided to look up solutions

online

and guess what

i was devastated the solution i found

online was so short that i felt like a

complete idiot that i couldn't think of

a more easier way

my solution definitely had to be the

most inefficient one

but now when i look back i realized that

the shortest solution is not exclusively

the most beautiful one

the path that we take might be somewhat

long but as long as we appreciate the

journey it surely is beautiful in itself

after all it is mathematics and all math

is clever and beautiful

so yes here's the puzzle

imagine you have four dogs and

they just happen to sit on the vertices

of a square whose side lengths are one

units

the dogs however as you can guess aren't

normal they are mathematically bred in

such a way that their speed is always

one units per second

and that the first dog always moves

towards the second dog the second one

always moves towards the third the third

towards the fourth and the fourth again

always moves towards the first one so

given all these wrong information if

they all began moving at once they

should meet each other at some point in

time so the question is what would their

paths look like and also what is the

distance covered by one of the dogs to

get there

now before proceeding i want you to

press pause and give this a try yourself

trust me the problem is quite rewarding

itself

so back to the puzzle

how do you even figure out the path of

such weird motion

will they run in circles spirals or

something else

for one thing they won't run in circles

no if their directions were always

oriented perpendicularly from the center

then they would have always run around

in circles but this isn't the case here

so they should slowly spiral in towards

the center right

but how do we show of it

well let's design a thought experiment

imagine that the dogs well their brains

are slow so they can only change their

directions after every t seconds

so at first the dogs will start moving

in their initial direction and only

after t seconds their slow brains will

suddenly discover that they are not

running in the right directions

so they'll fix their directions

instantly and start moving towards the

new directions again

and they keep on running in this manner

fixing their directions after every two

seconds

so running in this way gives us a sort

of spiral-like path for our dogs

but this is not the original path

because our original dogs should not

have any lag time at all they should not

fix their directions after a lag time of

t

they should fix it after a lag time of

zero that is they should move towards

the right directions instantly but in

this example where we chose a lag time

of t equals 0.2 seconds let's see what

happens to the path here when we keep on

decreasing the value of t so that it

almost reaches zero

i hope now this starts to make some

sense

yes the path that you see here is the

path our dogs take while moving in their

weird way but

hey this just became a whole lot

difficult how do we even measure the

length of this curved shape

is this even possible

well let's move back to our thought

experiment and explore it a little bit

further

here the dogs had a slow brain with t

seconds of lag time which meant they

could sense any changes only after t

seconds

now notice that a t seconds lag is the

same as calling it a t distant flag

because the dogs having a speed of one

units per second covers the same units

of distance in equally the same units of

time this means instead of saying the

dog sends any changes after t seconds we

can equally say that the dogs sense any

changes just after crossing t units of

distance

here we can see that after each step the

dogs form an intermediate square which

by the way is always smaller than the

previous one

let's name the sides of these squares

the initial one let's call it s1 and

then the next ones as s2 s3 and so on

ok enough notations let's talk business

how do we measure this curve distance

well that's where we start to use some

magic

no not literally we are not chanting

some spells to find out the answer but

what we're about to do at first might

seem a bit weird but the way things work

out at the end

it almost feels like magic

let's start building a graph

but hey we don't know any functions here

i mean it might have helped if we knew

the equation of the spirals at least we

don't even know that what graph are we

gonna make

well let's do a crazy thing we'll start

building a new one

which won't require knowing about the

equation of the spirals at all

again we start with the lag case

here after each step our dogs take we

put one t distance in the x axis which

is the distance covered in that step

and then let's put the side length of

the intermediate square formed after

that step on the y axis

and we keep on doing this for all the

steps

the graph that we see here is amazingly

useful for instance if you know the side

length of an intermediate square at

let's say this step

matching the side length here you'll

know that it took the dogs one two three

steps to reach this point so the

distance caused by them to reach this

step is this length in the x-axis

and this is true for any step if we just

know the side length at any step we can

find out the distance traveled by the

dogs to get to that step just by looking

at the x-coordinate

now this is actually a key discovery

take a moment to realize how this is

true

but how does this help to find the

distance travel to get to the center

well we already know the side length of

the intermediate square in the center

since it's the center the side length

has to be zero

but wait this means our answer is the x

coordinate for which that graph is equal

to zero

but how do we find where this graph is

equal to zero we don't even know the

function yet

okay but before that the graph you see

here is for the t distance lag case

where the dogs had a slow rain what

would the graph look like for the

original case when t approaches zero

this made me question the first time

could it really be that simple is this

graph in the original case actually a

straight line

well it certainly looks like one but how

to prove that it is one

this is the last clever thing we'll need

to solve this once and for all

in fact we don't even need to do much to

prove that it is indeed a straight line

if we can somehow show that the slope is

always the same that will easily prove

our claim because straight lines well

are straight so they don't change their

slopes at all

and finding the slope is relatively

simple for the lag case the slope for

any two consecutive steps would appear

to be the vertical change divided by the

horizontal change

here the horizontal change would be t of

course because we chose the length of

each step in x-axis to be t itself

in case of the vertical chain it will be

the difference of these lengths

let's call this s sub n so the other one

will be s sub n plus one

so the vertical distance becomes s sub n

plus one minus s sub n and the slope

should look like this

now observing our figure we can actually

find a relation between the two

consecutive squares so here if this

square side length is s sub n the next

one should be s sub n plus one

and we can spot a right angle triangle

here and in fact we can easily figure

out the length of the sides of this

triangle with what we already know

now using the pythagorean theorem we can

find the following relation

and since we didn't choose any two

particular squares this formula is

actually applicable for any two

consecutive squares in this figure

now remember our slope we'll use the

value of s sub n plus 1 here and the

slope now looks this way but this is the

slope for the lag case for the original

case the slope will be the limit of this

value when t approaches zero

i am not showing the calculations here

but it turns out this limit is

exactly equal to negative one which is a

constant

so the slope whichever squares you

consider is always a constant

so

there we have it this has got to be a

straight line

and since now we know the value of the

slope and we also know that this line

passes through the point 0 1 which

represents the very first square the dog

started with using these two information

we can figure out the equation of the

straight line quite easily

it turns out when this graph is zero the

x-coordinate is one which means the

length of the curved lines to which our

dogs traveled must also be one

this to me is amazingly beautiful that

the square we started with had the same

length that these curved paths have they

somehow happen to be equal this almost

feels paradoxical

so there we have it now we know our

answer

but this video was never about the

answer it was about the steps we took

that let us hear

and you might disagree but here's a

profound statement to conclude this

video it is better to solve one problem

five different ways than to solve five

problems one way

thanks for watching

and keep matting

you