What is Graham’s Number? (feat Ron Graham)

What is Graham’s Number? (feat Ron Graham)


>>BRADY HARAN: I think most Numberphiles are fascinated by the idea of Graham’s number, which is supposedly unimaginably big – but what actually is it? What does it count? Well, I think the best person to explain that is none other than the world famous mathematician himself, Ron Graham.>>RON GRAHAM: Suppose you took four vertices of a square – and, this is two dimensons, so it has how many vertices? Two squared (four vertices). Now, what I’m going to do is take these four vertices, and put – two more edges connecting all possible pairs of these vertices. Now how many possible edges would you be able to draw? Well, there’s something called a binomial coefficient, and there are four vertices, and you choose two of them, and this turns out to be six. So you see there are six edges here – all the ways you could join these four dots. And now, what we’re going to do – is – take the four dots again and look at each one of these possible six lines and color them red or blue in any way that you like, so for example – I might color this one red, and maybe this one red, and then I’d say, well ok, maybe I’ll color this one blue, and this one blue, and this one blue, and this one blue. Ok, so that’s one way, and you might say, well, okay, so what? I’m looking for certain special configurations – and, I’m not going to find it here in a square because this is too small; we need more dimensions. What about three dimensions? Well, what does a three-dimensional cube look like? Well, I can’t really draw it on a piece of paper, but you can kind of draw a version of it, namely, if you take your four vertices that form your square and then you kind of move it a little bit, well, that’s a picture you might recognize, how many vertices on these corner points here? Well, you can see there are eight, which is two cubed. So the three-dimensonal cube has – two cubed – eight vertices. 28 possible line segments. And if you color them, say, this one’s red, and this one’s red, and this one’s red, and that one’s red, but – you know – a lot of the others are blue, and what you’re trying to do is do it in a way that avoids something. Now, what is this thing you’re trying to avoid? That’s the whole point. You’re trying to avoid the following configuration: namely, four points in here someplace and all the six line segments have the same color (in this case it’s red) and, furthermore, these four points lie in a flat plane. They don’t stick up in three dimensions. Usually, if you took four points, you’d have three dimensions. It kind of spans a three-dimensional volume. Here, I want four points to be flat, and they form a square, and all six line segments are the same color. In three dimensions, you definitely can avoid such a configuration, because this only seems to happen on, just kind of the faces. And what you have to do – your challenge – is to go in a much larger dimension. And what do you mean by a cube – in say – four dimensions? Well, in four dimensions, you could take this picture, and translate it, and join the lines, but it gets kind of messy, but yeah. I’m not afraid; I think I can do that. Let’s see. Here’s a three-dimensional cube, and now I have to shift it, okay, well, I’ve gotta be a little careful here – and now I join the corresponding things. Well, you can see it’s going to be, as we can call it, a tesseract. Now how many different ways can you color these 120 different lines with red and blue? Well, with each one of them, there are two choices. So, it’s two times two times two times two 120 times, or 2 to the 120 (2^120). Already beyond what computers can do, and no computer can do 2^120 things right now. But the problem we’re trying to avoid having these four points that form a square lie in a plane and have all six line segments the same color and in four dimensions, you can do it. Not too hard to do. Well, what about the five-dimensional cube? In five you can avoid it, six – it takes more work, seven – yes, it takes a lot more work, the question is, can you always avoid it? The answer: no. If the dimension is large enough, you cannot avoid it. No matter how hard you try to color it, you can’t avoid it. And when is that dimension finally reached? Well, you know it’s reached by the time you get to a very large number, sometimes known as Graham’s number, a pretty big number, and you know for sure, in that dimension, you cannot avoid it. And the proof that you can’t avoid it is pretty complicated, pretty recursive, and that’s what causes this number to be so large. Now, you might say, well, what’s the truth? Sure, if you go as far as Graham’s number, in that dimension, you can’t avoid it, but maybe you can make it unavoidable in a smaller dimension. And it turns out that it appears that already in thirteen dimensions you can’t avoid it. Now, how could you check that by computer? Well, okay, no problem really, in thirteen dimensions, so “n” is thirteen dimensions, so how many vertices in your thirteen-dimensional cube? Well, 2^13 vertices, which is, what, that’s about 8192. Okay, a little over eight thousand vertices. Well, how many line segments? Well, 8192 times 8191 over two – you have to check – no matter how you color them with red or blue – you’ll make one of these things. Well, it’s uncheckable. You can’t check that; it’s impossible. You need to think! You can’t just compute. Right now, we don’t know. It’s between 13 – we know, maybe – uh, 12 is not enough, it turns out, and this other, slightly bigger number, which uh – I can – I can try to write it,>>BRADY HARAN: So there we go. As cubes move up the dimensions, from three dimensions, that I think we all understand, to four dimensions, to five – eventually we reach a number of dimensions in which one of these configurations must exist. And the upper bound on that, the point where it has to happen by, is this “Graham’s number”. Now, we know this number is unimaginably big, it was even in the Guinness Book of Records. But just how big is it? Well that deserves a video all of its own, and again, we’ve got Ron Graham to explain it. The links are on the screen here, I’ve also put them in the description under the video. So, why don’t you check it out? It’s a pretty big number.

100 thoughts on “What is Graham’s Number? (feat Ron Graham)

  • March 26, 2018 at 3:58 pm
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    Time travel paradoxes

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  • April 5, 2018 at 3:12 pm
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    What if you colored the vertices red and blue as well? What's the fewest number of dimensions where it's impossible not to have a 2-D square of vertices with all red or all blue lines and vertices?

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  • April 6, 2018 at 8:22 pm
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    I don't understand why 2 to the power 33,550,336 posibilities need to be checked for 13 dimensions. Doesn't one configuration whereby the 6 same-colored vertices does not occur suffice? Or do these valid configurations become very scarce with higher dimensions?

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  • April 19, 2018 at 1:28 am
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    The reason that it does not matter if you are using plank times or millennia is not actually just because the numbers are big. It is because, the way the numbers are written, a third significant figure would alter the result many orders of magnitude more than any unit conversion could. The calculation is not precise enough to make units relevant. To demonstrate, type 10^10^1.5 and compare it to 10^10^1.52. Notice that those small significant figures add two orders of magnitude to the answer! In the chain written in the video, the difference between 1.1 and 1.11 would be huge enough to make units irrelevant.

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  • April 20, 2018 at 12:29 am
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    On a scale of 13 to Graham’s number…

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  • April 27, 2018 at 1:14 am
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    whats the application for this?

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  • April 30, 2018 at 5:36 pm
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    Quick mathzzz

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  • May 6, 2018 at 6:56 am
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    Is he alive today?

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  • May 15, 2018 at 1:01 pm
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    Am I ignorant because I wonder, if you cut the in diagonal halves, you would have four points with six egdes on one plane, as stated to avoid? I ask because there would be six of these possible planes on any cube, two for each axis. Or are they not counted because these would not be square?

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  • May 20, 2018 at 5:11 am
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    He talked about coloring the lines red and blue, but Numberphile colored them red and cyan. That just ruined it for me.

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  • May 23, 2018 at 4:46 pm
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    So what are the applications of knowing this information? How exactly is it that this multidimensional "cube" HAS to be completely one color and it is completely unavoidable?

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  • June 10, 2018 at 2:50 pm
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    Its just a little larger than 13

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  • June 13, 2018 at 7:48 am
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    He is trying to make a surface out of edges which is huge somewhat to the order 13 because 5 plus something and that something is Phi. .

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  • June 25, 2018 at 8:45 pm
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    G64 + 1 BOOM, I'm better

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  • June 27, 2018 at 6:02 pm
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    WHAT MAKES THE LOUDEST SOUND?

    GRAHAM('S) BELL

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  • July 6, 2018 at 9:14 pm
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    Harrison Wells has taken another body

    Reply
  • July 9, 2018 at 3:03 pm
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    Base is ten extra dimensions are two so twelve. So twelve is possible Phi × 4 is the max. Four is the constraint.

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  • July 17, 2018 at 1:41 am
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    Instructions not clear, mind permanently disappeared

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  • July 19, 2018 at 5:27 am
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    This guy gives me hope in mankind….

    Reply
  • July 20, 2018 at 2:00 am
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    IMPORTANT: The upper bound is not Graham's Number anymore, and, strangely, NEVER WAS. The original bound was proposed in 1970 and was about G(12) in terms of Graham's function. Only in 1977 was the paper about G(64) published. In terms of up-arrows, the new upper-bound is 2^^2^^2^^9, which is still REALLY big, but it's actually less than even G(1).

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  • July 21, 2018 at 10:24 pm
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    Aham's number

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  • July 24, 2018 at 5:30 am
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    Three

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  • July 25, 2018 at 1:24 pm
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    13d—125375757748856786788789707896958967787

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  • August 5, 2018 at 8:31 pm
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    a+1=counting
    a+b=addition
    a×b=multiplication
    a^b=exponentiation
    a^^b=tetration
    a^^^b=penetration
    a^^^^b=intercourse
    a^^^^^b=sigulation

    Reply
  • August 5, 2018 at 10:54 pm
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    How many possible lines of chess moves can be played from the first move onward?

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  • August 20, 2018 at 3:25 am
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    I’m confused.. why is the pattern that their trying to avoid not possible to avoid at some point?

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  • September 1, 2018 at 2:56 pm
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    DAMN!! my mind is dying

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  • September 1, 2018 at 9:11 pm
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    ok
    THIS is epic

    Reply
  • September 16, 2018 at 12:52 am
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    Graham's number + 1

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  • September 22, 2018 at 9:38 am
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    It is awkward to know the graham number Is Explained by Graham itself who discovered it. Lolz

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  • September 25, 2018 at 9:51 pm
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    The biggest number is 9

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  • October 9, 2018 at 1:00 am
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    So what's the proof here? Graham's number is where it's unavoidable says what?

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  • October 10, 2018 at 6:49 pm
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    how about if you made all the diagonal edges red and the strait (in x,y,z,…) edges blue??
    wouldnt that avoid having any squares of all one color??

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  • October 18, 2018 at 7:49 pm
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    Wait doesn’t the newest version of string theory call for 12 spatial dimensions and it said 13 may be unavoidable for crossing lines coincidence probably but SCIENCE

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  • October 20, 2018 at 9:22 pm
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    So what?

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  • October 21, 2018 at 1:40 pm
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    0:57 tho

    who else cringed

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  • October 28, 2018 at 1:08 am
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    Couldn't you clone using this?

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  • November 1, 2018 at 3:09 am
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    3:32 That's from Wikipedia but with the colors reversed, right? A shame you can't even make your own rotating tesseract.

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  • November 8, 2018 at 10:55 am
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    I avoid this

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  • November 12, 2018 at 7:51 pm
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    Imagine it'd be 42d. WW3 would start

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  • November 14, 2018 at 11:17 pm
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    I love his voice and want more videos narrated by Ron Graham.

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  • November 15, 2018 at 1:56 am
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    Huh, he's still alive… That's unusual for having something named after you

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  • November 15, 2018 at 2:01 am
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    They say that at dimension (Graham's number)^Graham's number it can be avoided, but only if you ate a banana in the last 24 hours

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  • November 18, 2018 at 4:50 pm
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    this sounds an awful lot similar to the ramsey theory graphs, is there some correlation?

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  • November 19, 2018 at 5:33 am
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    Grandma's number…

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  • November 21, 2018 at 7:21 pm
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    3:28 it s magic

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  • November 22, 2018 at 4:16 pm
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    Well what if 13 dimensions is enough to make the 1-colored plane unavoidable? Would Graham’s number be useless then

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  • November 30, 2018 at 4:25 am
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    Why can't you just replace one of the center crossing lines with the other one?

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  • December 6, 2018 at 5:07 am
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    GP means googolplex

    GP^^GP = F1

    GP^^…F1 number of arrows…^^GP = F2

    GP^^…F2 number of arrows…^^GP = F3

    F100 = GP(F99^)GP

    Surely this beats Graham's number.

    Reply
  • December 21, 2018 at 7:40 am
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    THIS VIDEO IS MY TOP 5 ALL TIME FAVORITE YOUTUBE VIDEOS SINCE I FOUND YOUTUBE IN 2006!

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  • January 16, 2019 at 11:22 pm
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    Can someone please explain me how did Mr. Graham able to conclude this "If the dimension is large enough you cannot avoid same color vertices configuration". Is this merely a hypothetical assumption or do we have a concrete evidence of this? Thx in advance. 🙂

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  • January 30, 2019 at 12:47 am
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    I just don't know why the they don't teach us this kn School.
    School makes us hate everything….

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  • January 31, 2019 at 7:04 pm
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    With what respect(according to what) is he colouring the lines(edges) plz reply??????

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  • February 2, 2019 at 9:13 am
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    {4,64,1,2} is bigger than g_64

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  • February 7, 2019 at 12:46 am
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    Well if Graham found it then how come he doesn't know it??

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  • February 21, 2019 at 5:37 pm
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    What if I colour ALL the lines blue except ONE which is red? Haven't I avoided it? There was no rule of using one colour multiple time

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  • February 26, 2019 at 11:31 pm
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    His hands moves via sine-wave function.

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  • March 9, 2019 at 12:02 am
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    So is the 14th dimension just completely made up of these uni colored squares?

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  • March 9, 2019 at 4:14 am
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    What's interesting to me is not how big the number is, that's just a
    number, it can always be bigger. What's interesting is the application.

    1) how would a 14 dimension cube even look like??

    2) how is he so sure it's unavoidable????

    3) what exactly triggers AVOIDABILITY on the 13th and 14th ????

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  • March 15, 2019 at 6:09 pm
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    at least at Grahams number one can't avoid that kind of configuration. But how does he know?

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  • March 16, 2019 at 5:33 am
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    Love videos on Graham’s number. Graham is a genius!

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  • March 17, 2019 at 9:45 pm
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    Wait, so it's not avoidable in the 13 dimensions. Is that why string theorists say, there's only 10-11 dimensions, because everything would collapse in the 13th dimension? Or is that just a fishy coincidence ?

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  • March 19, 2019 at 4:26 pm
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    my man is shakiiiiiiiiiiiiiiiiing

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  • March 19, 2019 at 6:02 pm
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    This guys voice reminds me of Bob Ross for some reason

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  • March 23, 2019 at 7:04 pm
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    I have trust issues with people who have an unsolved Rubik's cube.

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  • March 28, 2019 at 4:49 am
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    Would love to know why it is so big, and what is the reason for the recursion. The only thing I noticed is (4 choose 2) is 6, and 2^6 is 64, and G64 is Grahams Number. What is the connection?

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  • April 3, 2019 at 9:55 am
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    Honestly I think mathematicians just dream up this BS because they have nothing constructive while all the engineers are making space engines, biologists are curing cancer and chemists are making hangover free alcohol or some such things…

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  • April 6, 2019 at 12:15 am
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    ( 2^(x-1) ) * ( (2^x) -1 )
    where x is the number of dimensions, represents how many lines will be created in the given dimension.

    I started with this formula:

    ( 2^( 2x-1) ) – ( 2^x )

    Using algebraic manipulation I ended up with the former. On paper, at least, this first formula is much simpler than what I started with, but the second formula is more tangible to perceive.

    Reply
  • April 6, 2019 at 9:23 pm
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    Why didn't numberphile explain this in their first video about grahams number?

    Reply
  • April 19, 2019 at 9:12 am
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    Why is his hand so unstable

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  • April 19, 2019 at 7:56 pm
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    Whats this thing that they have about writing on brown paper??

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  • April 25, 2019 at 7:36 pm
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    Why select such a conservative Progressor as 64 ???????? G65 is so much larger

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  • May 11, 2019 at 5:17 pm
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    so graham's number is an upper bound to how many dimensions must a complete 2-colored graph of a cube have to be forced to contain a complete single-colored coplanar 2-dimensional graph of a cube. what if the graph could be 3-colored? 4-colored? graham's number-colored? or what if it could have as many colors as dimensions? what if the single-colored graph we're trying to avoid was 3-dimensional, not 2-dimensional? 4D? graham's number-dimensional? we could make a function from this: f(c, d), where c is the number of colors of the main graph and d is the number of dimensions of the subgraph. we can also put countable ordinals as c, where if n is the number of dimensions in the main graph, then c becomes the nth member of the fundamental sequence of the ordinal, so f(omega, 2) means that the graph could have as many colors as it has dimensions

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  • May 12, 2019 at 12:10 pm
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    2^33550336 easy

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  • June 1, 2019 at 6:36 pm
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    5D, 496 possible edges.

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  • June 4, 2019 at 8:21 pm
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    I got Graham's Number tattooed on my ankle today. Original instead of an infinity tattoo 😂😂

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  • June 20, 2019 at 7:18 pm
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    What we avoiding again ?
    I couldn't understand it

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  • June 22, 2019 at 3:27 am
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    3:48 " far beyond what computers can calculate" I did the calculator, and it did mate

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  • July 6, 2019 at 1:00 am
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    Wtf I'm been watching this everyday for the past 5 days.

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  • July 19, 2019 at 2:24 pm
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    2^120 equals 100000000….(120 zeros) on base 2

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  • August 13, 2019 at 10:26 am
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    I confess – I prefer Numberfile's videos about numbers I can get a handle on!

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  • August 15, 2019 at 6:41 pm
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    Very cogent explanation by Ron Graham.

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  • August 29, 2019 at 4:16 pm
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    3:28 Hey, I'm on the video !

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  • September 7, 2019 at 3:44 am
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    What happens if you call the Ackerman function with Graham's number as the arguments?

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  • September 18, 2019 at 8:27 pm
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    What did i miss? Wich config you have to avoid? Same color in a side or differnt colors in.a.side?

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  • September 19, 2019 at 2:09 am
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    The best person to explain Graham's number is day[9]

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  • September 24, 2019 at 11:11 am
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    So is grahams number the dimension or that ways to color the lines????

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  • October 4, 2019 at 12:47 am
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    It like watching Bob Ross teach Math.

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  • October 5, 2019 at 6:26 pm
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    what is the purpose of finding those specific configurations?

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  • October 22, 2019 at 2:15 pm
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    World famous mathematician but can't draw a straight line….

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  • October 27, 2019 at 10:06 am
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    ''Already beyond what computers can do.'' And then they show the number. Hahaha!

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  • October 27, 2019 at 4:34 pm
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    This other, slightly bigger number.

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  • October 29, 2019 at 2:29 am
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    Basically it is known for being the worst upper bound in history.

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  • October 31, 2019 at 1:27 am
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    So the upper bound on the number of dimensional cube where a plane with a square that is all one color is unavoidable is somewhere between 13 and graham's number. Physicists: "the cosmological constant problem is a theoretical prediction that differs from observation by a factor of 10^120. It is the worst prediction of the worst precision in physics." Mathematicians: "Hold my beer."

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  • November 3, 2019 at 7:25 am
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    Sometimes maths just seems so pointless

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  • November 6, 2019 at 3:51 pm
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    2019 from Still Untitled on Tested

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  • November 7, 2019 at 6:27 am
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    It just got over my head 😴

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  • November 19, 2019 at 5:13 am
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    3:06–3:10 aaahaha, Yupp😂😂😂😂

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  • November 19, 2019 at 3:44 pm
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    wow he has the most wobbly straight lines ever

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  • November 24, 2019 at 12:11 pm
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    5:59 famous last words

    Reply

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