Friends and Strangers Theorem - Numberphile
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- เผยแพร่เมื่อ 1 ส.ค. 2024
- A famous mathematical theorem explained with the help of Facebook, One Direction and a collection of Numberphile's Facebook followers.
More links & stuff in full description below ↓↓↓
Videos features Simon Pampena - / mathemaniac
NOTE: Yes we should have used greater/equals symbols at around 5:15, but the meaning pretty clear and Brady and Simon live about 10,500 miles apart --- so not worth a re-shoot!
More on the theorem: en.wikipedia.org/wiki/Theorem_...
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Yes we should have used greater/equals symbols at around 5:15, but the meaning pretty clear and Brady and Simon live about 10,500 miles apart --- so not worth a re-shoot!
hi
Naa, it was actually fun, the people, even your fans are idiots... 3 is just a constant, yes, it's not that interesting, but is not always gonna be a super algorithm that creates planets -_-
Reply becasue else it is red line and i will be sad 😔 😂
Numberphile who’s the person who writes TH-cam comments? Just asking...
The whole "3 friends or 3 not friends" thing is just like the riddle "How far can you go into a woods? Half way because then you are walking out of it."
Don't round it up, Brady.
Don't ever round it up.
FFS Brady! DO NOT ROUND IT UP!!!!
Note to self: avoid threesomes at all costs.
"There are more than six people on facebook. You know that, right?"
I mean, there must be at least 10 if you round up.
and today we learned, that 3 > 3 = true
yep, he said "at least" so he should have written ≥3 instead
minauras yeah I know, I was just referring to what he wrote :3
Brandan09997 not if he takes it lightly
Brandan09997 and you're a piece of shit :3
and i learned that
(3 > 3 = true) = true
Three people who are my mom or my dad? That does make for one very awkward christmas party.
It is time we tell you about the flowers, the bees and the Turtle.
You'd be the third person, mate.
Cheer up mate, LOTS of presents :)
Psykodamber.dk Surely you mean tortoise, bees don't swim very well ;-)
1:53 😂😂 I love how much offense he takes to the approximation
The one direction example of the connections would now have 5 red lines 😂
TOO SOON
@@alexsawyer8467 what do you mean, you commented 3 years later hahah
A popular application of the pigeonholes theorem. Well explained!
can you please explain?
@@erek _"And ever since, Louis-Math had not explained"_
@@fantiscious lol
@@erek pigenhole principal is basically if there are more numbers of pigeon than hole than one hole should have more than one pigeon. So n>m where n is number of pigeon and m is number of holes than one hole will fundamentally have more than one pigeon it's simple. Lol
@@arpitdhukia9026 i know what it is. How is that related to this video?
I kept waiting for there to be more to this one. It was just so intuitive I was waiting for another step that would make me think. I guess they can't all be winners.
I have a maths exams next week and this is surprisingly relevant to the course. This is actually a question that can come up. Now this counts as part of my revision :D
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me too,
combinatorics for computer science
i got a test in a week, this can totally be one of the questions!
I'm doing the D1 paper too! This exact question actually came up on a mock I did yesterday!
what the hell is albert neville is talking about?
Albert Neville Chill out you worthless goat.
This is simply amazing, both the concept and your explanation
Shouldnt be ">=" instead of ">"?
Good thing I have no friends
This is my favourite problem! I am so excited that they have done this!
Sometimes if find myself wondering what application some of these mathematical principals have. Then, I'll either do the research or accept the fact that sometimes it's simply about finding order out of chaos. Thanks guys!
Simon is one of my favourite people in these videos.
This video is an icebreaker! Tried the whole day to understand Ramsey's Theory from my university literature, without any succes. Thanks to this video I understand everything. Thanks!
That graphic at the end showing how you inadvertently made a group of strangers by trying not to make a group of friends makes it look like it's possible to do this with 4 or 5 people. I had to actually draw my own diagram with 5 people to reaffirm what you showed earlier with it mattering how many connections they each have.
Since when is 3 > 3?
Cuz no one wants 3
Simon is just amazing :) Great video!
Great video! Do more about this!
You can tell this video is old b/c it features a positive reference of Facebook
"There is exactly 32,768 different ways...don't round it up" hilarious!
I saw a question where you had to explain this in a D1 A Level maths paper (which I'm taking on Tuesday).
Great explanation, really!
great explanation! thank you!
That's pretty awesome! Thank you
This entire video sounded simple as he explained it. Intuitive, even. So I tried to derive the conclusion on my own, starting with conclusions one can make about groups of three people, then four people, and hey, why not seven...
By the time I got to ten my head was fried. I'm beginning to see why Graham's Number is used in Ramsey Theory.
Well Explained. Thank You
This is such an amazing way to learn math at master's level.
Hi Brady , I know you work very hard doing these videos and I´m sure you take the time to check every single one of them to deliver them the way you want , but, I´m from argentina and do enjoy your videos a lot only when they come with the sound. I´ve seen literaly hundred of videos and I´m sorry to tell you that only half of them have sound in all your channels , sixty simbols , periodic videos , veritasium, etc.
unfortunately this is one of those videos without sound and I can´t let it pass
thanks for all your good work, I learn new things every day with this videos, cheers
This reminds me of when someone said to me that no person is more than 6 relationships away from any other person.
What about higher numbers?
If you have more friends, does the minimum number of people in a friendship group increase? Or is it always 3?
If you have 8 people does it change to 4? Or was the fact that 3=6/2 a coincidence?
Brady, thanks for giving me Simon Pampena on TH-cam. That makes his appearances on Outrageous Acts of Science more exciting to me. (And Matt Parker too.)
I like to think this might be laying the ground work for the awesome upcoming Graham's Number videos!
...I _like_ to think this, but I have a very weak grasp of the problem that Graham's Number is the solution for. But it involves 2 colours :D
Yeah, I thought that too :D Then after a quick google it turns out they're both part of Ramsey Theory, so I think that's quite probable!
They did a Graham's Number video already: v=XTeJ64KD5cg
***** Can't quite say they did a very good job of explaining the problem that led to Graham's number, though. Would be nice to seem them return to it.
Guess what...
great video
Now I feel like adding Simon Pampena up on Facebook, just because of his hilarious reaction when he says he doesn't want to be friends with anyone.
I once read that there is a way to calculate that you know everyone on earth within like 15 corners. So with only around 15 relations you are connected to everyone on earth. An explaining video to that would be really cool :) Numberphile
Its popular name is ramsey problem. R(n,m) is the minimum number k such that any red and blue coloring on Ck (complete graph of k vertices) always contains Cn red or Cm blue. Here, R(3,3)=6. This problem is so hard as Erdos said like "if alien invade us and they give us option to answer R(6,6) or war, it's better to choose war"
i'm just waiting for one video with this guy in it i'm actually going to enjoy watching.
maybe next time...
I lawled so hard when he said: Dont round it up. He was so serious:DDD
His sass towards Brady is the bessst
Thought he was gonna talk about the "At least through 7 strangers, you'll meet someone you know" thing.
It is strange... I saw many of these things in math class back in high school. Back then I assumed it was normal but it seems like I just had a really really awesome math teacher.
Simon is awesome!
"There are exactly 32,768 different ways to do that, Brady. Don't round it up." LOL
IMHO though the "triangles" should have been introduced right in the beginning in order to make even more clear what was meant with three people all being or not being friends.
Exactly, in the first drawing, they show triangles, and then when Simon explains it with the table, he just shows the possible combinations out of 5 connections. But these combinations do not necessarily form triangles because you don't know what the other guys are doing. And then he shows the proof. To me it seems like the middle part does nothing to aid in proving it, they might as well have left that out. Still a cool video though.
I think it's pretty self explanatory.
@TijnvanBoekel follow the video more closely, he needs the middle part to prove that for any one person, the minimum number of friends or not friends is three. Only because of this can he say that in all situations is there a triangle.
If he didn't do the middle bit consider this - I can say "what if the first person (with three friends) were to have only two friends with those people then we can avoid the triangle. I.e. Turn the third line red". I agree it is obvious this is just displacing the problem, but if he is being mathematically rigorous (despite 3 > 3) then he should prove it.
Malkitasoman that makes sense, thanks :)
bloody triangularists
3 people that are my mom or my dad? =O Great vid as always Brady and Simon :)
🤯 that I randomly watch this video 7 years after it’s been uploaded and I literally know the couple whose photo was used. Lol. Friends and strangers for real.
MIND=BLOWN.
"Sorry I don't want to be friends with you don't try it and just stop requesting to be friends, I don't want to be friends with you ok? No." lol
I watched this whole video wondering what the point of this video was until the end where it all made sense.
Question for Simon, what do you do in your spare time, and how easy is it for you to think and see the problems that you are explaining. It would be fascinating to think like a mathematician, could you describe?
I have a strong penchant and absorption for mathematics, and, reading the comments, should this build into something larger and more foreign to myself (it appears the Ramsey Theorem is what this builds into), it would greatly satiate my thirst for being able to better understand maths when I cannot afford to take classes on it all.
Thank you, Brady and Simon! :)
For anyone interested in this topic this comes from an area of mathematics known as Ramsey Theory. In Ramsey Theory we ask ourselves, "How big must a system be before we can always find a certain pattern?" The problem shown here is the most classical example and is actually just a simple example of a much broader theorem known as Ramsey's Theorem. I invite all the intrepid minds to look a little deeper into the subject. You may enjoy what you find.
This has to do with a fabulously mysterious area of mathematics called Ramsey numbers
There is a result in graph theory that states that if G is a graph on 6 vertices, then either G contains a triangle or G'(complement of G) contains a triangle.
"There's exactly 32,768 different ways you can do that, Brady. Don't round it up."
There's also this result which says there will always be an even number of people that are friens with an odd number of people.
Isn't it easier to just say 6(people) / 2(choices) = 3(minimum of one choice). If you just take six and split it in half, you get 3. If you take a smaller group (any 2 or 1), then the opposite group becomes larger than 3. Thus, with 8 people and 2 choices (friends or not), at least 4 people would know each other or not. With 10 people, at least one of the two groups would be made up of 5 or more people.
Thanks!
Wow, this exact question came up on my D1 mock, that's a crazy coincidence.
I like how you try to make it relevant and then use the Addams family for the diagram...
amazing !! 😍
Just in case anyone else was wondering how he gets the 15:
With 6 people:
Person 1 can have 5 unique connections. (Person 2, 3, 4, 5, 6)
Person 2 can have 4 unique connections. (Person 3, 4, 5, 6)
Person 3 can have 3 unique connections. (Person 4, 5, 6)
Person 4 can have 2 unique connections. (Person 5, 6)
Person 5 can have 1 unique connections. (Person 6)
Person 6 can have 0 unique connections.
5+4+3+2+1 = 15
This can be mathematically modeled as: .5(n^2 - n) or .5n(n-1)
In my field we refer to this at Metcalfe's Law (specifically referring to telecommunications)
I love his tone
I have such great memory to the Adams Family movie. There was no sequential movie because the male husband actor died, see wikipedia. I would have loved to see more Adams family movies, but the children don't stay young.
A number of commenters are saying that this is obvious because if less than half the group are friends, more than half will be strangers and vice versa. This is not true because you cannot neatly classify a whole group of people into friends and strangers. You may have two friends who don't know each other, and two people you don't know may be friends.
In a group of eight, for example, you may say that there must be at least one group of four that all know each other or are all strangers to each other. That is not true. Imagine (or rather try graphing it out because it is probably hard to visualize) eight people, numbered one to eight. In this group, two people know each other if they're separated by one or two, otherwise they're strangers. Also, we're using modular arithmetic, so 1 and 8 know each other, and so do 1 and 7. In this group of eight there is no group of four people who all know each other and there is no group of four people who are all strangers. (If you find any, please let me know.)
I hope this helps.
≥
Can you guys do a video about essential singularities in complex analysis and Picard's Theorem? It's my single favorite Theorem just because of how awesome it is to imagine. Thanks :)
Wednesday would be a scary friend to have on facebook. You'd say something similar to, "My grandma passed away..." and then there would be a singular 'like'...
Very interesting
Best phrase ever "there's more than 6 people on Facebook"
Fun fact: Graham's Number, the former largest number ever used in a mathematical proof, actually stems from this fascinating 'order out of chaos' theory, also known as Ramsey theory!
I was wondering if he was going to show the formula for determining the number of possible connections, which is (n^2 - n) / 2. That is how you get 15 connections from 6 points (without having to count each line).
It is worth noting that in a group of six there can be a situation were there are no 'triangles' of friends/not friends, but in those cases there is a 'loop' of 4 people which does count for this theorem. In those loops, however, the people across from each other are of a different relation than what makes up the loop. Easy example: take people {1,2,3,4,5,6} with friend connections between 1-2 2-4 4-5 5-1 3-6 and fill the rest with not friends. Here you have a loop of friends in 1-2-4-5 and not friends a couple of ways but 0 triangles of either type.
Not true. {1, 3, 4} forms a triangle of non-friendship.
Nice idea.
If you want to see a generalization of this, look at edge colourings on regular complete graphs.
I wouldn't be connected to my family either if they looked like that.
By being friends with two people and the strangers with two, the remaining connection forces the trio of friends or strangers.
Simon rocks, quite frankly
"Don't round up Brady" Best damn part of the whole video.
The question then leads to why 3? In a group of 6, you always have at least one trio that are all green or all red. Does this hold true for higher group sizes? In a group of 20 people, is it still trio, is it a tensome, is it a different size, etc?
is about ramsey's theorem, R(3,3) is 6 ,r(4,4) is 18(means in a 18 people group ,there will always be a square,"four-some") and thats it , we dont know how many people is needed for pentagon(or "five-some)
The first person can has exactly 5 connections with the 5 other people, obviously. The second person has 5 connections as well, but one of those connections is with the first person, which we don't need to count twice, so we get 4. The third person has 5 connections, 2 of which we counted already, and so on. By the time we get to the sixth person, there are no more connections left that we didn't count yet, so the number of connections between 6 people is 5+4+3+2+1=15.
2^15 derivation is simple: there are 15 "connection" lines, and each has a possibility of 2 states (friends or not friends). If you want to think of it like a tree diagram, connection 1 has two states, and from each of connection 1's states connection 2 has two states, and so on (2 x 2 x 2....), or 2^15 for short because there are 15 connections.
"A Threesome of Anonymity"
Simon was sassy in this one. I like it.
"You can always find three people, that are [...] your mom and dad."
That would be strange... o.o
What about people who quite facebook entirely? If you have a group of 6 people who quit facebook, is it possible to have less than 3 of those people either be friends or not be friends? Cuz i quit it, and i know a large number of people who have as well.
There are fifteen possible lines between six people. But for each line we have two options: friend and not friend. So we have a total of 2x2x2.....x2 =2^15 ways of obtaining all possible combinations.
It gets even better: every combination in a six person's game had 3 connected people by the same kind of line, red or green, then does it mean that for a N person's game we get N/2 people are always connected by the same kind of line? I think so, but haven't been able to show it.
Why wasn't the term Ramsey Number mentioned? In the video you showed that R(3,3) = 6. "Paul Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5,5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6,6). In that case, he believes, we should attempt to destroy the aliens."
snappas Yeah, they really missed over the opportunity to introduce the idea of Ramsey theory.
you should tell us if there is a rule that allows us to know how this works in smaller or bigger groups, like what happens with 7 people? does it form a square or what?
Don't round it up ... thank you, Simon!
His face when he said "don't round it up"
So how big a party would you need to guarantee that there was a group of 6 mutual friends or a group of 6 mutual strangers?
I feel like I should know this; what's the math to get 2 to the power 15? So the math to get that it's power 15.
Now that I've typed this up ... is it because 5+4+3+2+1=15? Right? (Start with one person who has 5 people that they can be connected to in one of two ways. Move to the second person who has 4 people remaining that can be connected to in one of two ways. Etcetera.)