[0:00] Are you ready for the biggest number I believe 
has ever been used in a game? And the way it's  
[0:05] generated, I think is like super interesting. 
It happens to come out of the game of Magic  
[0:11] the Gathering, which is why I have all these 
cards around me. Now, I don't play Magic. Um,  
[0:15] friends of mine do, who are massive dorks, and 
they assure me, and they're in the room right  
[0:20] now. They assure me it's a lot of fun, and I get 
it. There's lots of cards. The cards interact,  
[0:25] which is how we get this ridiculous number. 
You got some uh like creatures. So, here I've  
[0:30] got a worm spelled incorrectly. And then you 
got spells you can play that change things in  
[0:35] the game because it's like a two-player 
game against someone else. Uh strangling  
[0:38] soot. Don't know what that is. Uh and Armageddon, 
destroy all lands. There you are. I'll play that.
[0:51] I regret calling them dorks.
[1:00] This video brought to you by Jane Street and their 
brand new neural network puzzle. Details later,  
[1:08] we're going to need somewhere new to film. 
Right, new location. Now, the thing about Magic  
[1:13] the Gathering is it's been around for now over a 
third of a century. And if you're thinking that's  
[1:18] not right, Matt, it came out in 1993. I have some 
terrible news about 1993 for you. But the point is  
[1:26] it started well it was the brainchild of someone 
with a PhD in mathematics and it's only got more  
[1:32] complicated since and over the three decades 
since then every possible edge case has come  
[1:38] up. There are now almost 30,000 different cards. 
And so the way they interact has become just an  
[1:46] unwieldy level of complex. And for that reason, 
they've had to introduce some rules to stop things  
[1:53] from getting too out of hand. Because a lot of the 
cards involve calculations and numbers, regulation  
[1:58] 107.1 states that only whole numbers are allowed, 
only integers. Let's try and keep this nice and  
[2:08] simple. If it's something you can't calculate 
using integers, well, regulation 107.2 says,  
[2:14] tough. If you can't calculate it, you get a zero. 
We're keeping it neat. Now, once you get to loads  
[2:21] of cars interacting, you think, well, hang on. 
What if there's like a runaway feedback loop?  
[2:26] Well, regulation 104.4B says no to that. If you've 
got some kind of, you know, feedback cycle, which  
[2:35] is going to carry on infinitely long, no deal. 
The game is officially a draw. And that's assuming  
[2:43] it's a cycle where no player can stop it. If a 
player can stop it, then that's fine. and they can  
[2:47] choose some some arbitrary level at which at which 
they quit. And so within those rules, in theory,  
[2:54] you can only play moves which will give you a 
manageable end result. And that worked. Everything  
[3:00] was nice and manageable for a very long time 
until someone found a loophole somewhere above  
[3:06] big numbers, but before infinity. Our ridiculous 
combination involves these three cards which we  
[3:14] have managed to get our hands on the original. 
However, for us non-magic the gathering people,  
[3:21] h they are dense. There's a lot of like wonderful 
artwork. There's a lot of information on these.  
[3:26] And so what we've done is we've made three 
simplified standup maths versions and we're  
[3:32] going to use those to explain how it all works. 
Starting uh with this one, card A, double it. Now,  
[3:41] this card doubles the number of tokens that are 
put into play. And the one nuance here is you'll  
[3:46] have like your original creature cards or like you 
know, spell cards. And you can also have copies of  
[3:52] them that are called tokens. And the game deals 
with them subtly differently. The original and  
[3:57] all the token copies. This is if you're putting in 
token copies of something, they get doubled. Now,  
[4:02] we'll get to card B in a moment because before 
you play B, you have to have card C in action. And  
[4:07] card C says once you do play B, you're going to 
get another token copy of it appearing. And once  
[4:13] you've got A and C ready, our ridiculous loop is 
set to kick off. Now, it's time for card B, which  
[4:20] in the original is called Astral Dragon. We've 
just called make two because when you play this,  
[4:27] you make two token copies of A. So, in theory, 
we need two copies of this to now enter the game.  
[4:34] But because they're tokens entering the game, 
card A's power is to take entering tokens and  
[4:39] double them. So, we'd actually get four. So, 
just by playing this one card, we should get  
[4:44] four token copies of A appearing as if by magic, 
the gathering. Now, to get those copies, well,  
[4:53] okay, this card over here, I know I'm going 
a little off. Um, this mystical teachings  
[4:58] card says I can search search your library. So, 
I'm going to use that to head to the library.
[5:12] So, at my local library, I was able to make some 
extra photo copies of card A, which if I now cut  
[5:19] them apart, gives us four token. And you can tell 
they're token cuz they're now grayscale copies of  
[5:28] of A. So there we are. We can put them all into 
play. I'll just put them down here. We're now  
[5:32] going to resolve the C issue, which is by playing 
B, we also get an extra token copy. And I've  
[5:39] got a couple down here. I've made these a little 
smaller. You'll see why in a moment. So we put one  
[5:43] token copy in of B because of C. But look at this. 
There's five A's over here. And they all see this  
[5:50] happening. And each one of them, well, the first 
one here decides that it's going to double this  
[5:56] one coming in. Okay, so now we got two of those. 
And then the next one sees both of these coming  
[6:00] in, so it doubles those. And now we've got four 
of them. And then the next one sees these four  
[6:05] and doubles them. Then we get eight. And well, 
there's five of them. So we have to double it five  
[6:10] times. So we get two to the power of five token 
copies of B. There are now 32 extra B's in play.  
[6:19] We've just put in 32 newbies. And every single 
one of these tokens says, "When this enters play,  
[6:24] put two token copies of A in." So, I need to 
add another two token copies of A. But guess  
[6:29] what's over here? Already five A's. And they're 
going to do the same thing to the incoming two  
[6:35] A's I did to the incoming 1B. Now, a single 
B got doubled five times. We had 32 of them.  
[6:41] If we have two A's coming in, it's going to 
get doubled five times, which would be 64.  
[6:49] So, here we have this is 64 A's, which 
I'm now going to add in to our A pile. Oh,  
[6:59] no. Because if I put these in, we now have a total 
of 69 A's over here. And we've only deployed one.  
[7:08] We've done that one. We've done one of our 32 Bs. 
So now the next the next B comes in. It's going to  
[7:20] make two new A's and they're going to be seen by 
69 pre-existing A's. So it's the two is going to  
[7:29] get doubled 69 times. Wow. We're about we're going 
to have to add two to the 70 A's. And here we go.  
[7:42] So, I have put these um we're now running 64 to 
a sheet and we have an additional 2 to the 70  
[7:50] copies of A plus the 69 that were already here. 
How many do we have? Could we work it out? I've  
[7:56] worked it out. Is that many? Look at that. It's 
a number with 22 digits. Oh, little confession.  
[8:04] I didn't photocopy them all cuz if we wanted 
to have This is basically 10 to the 21. And  
[8:09] if we wanted that many card A's at this scale, the 
amount of paper it would take would fill more than  
[8:16] one entire shipping container with just paper. In 
fact, it would fill multiple shipping containers.  
[8:24] It would fill enough shipping containers to 
fill 1 million shipping container tankers. Uh,  
[8:31] that's a lot of paper. We'll talk more about 
outrageously big numbers in a moment. But just to  
[8:35] complete this journey, we're going to keep going 
through B's. And each time we go through B's,  
[8:41] we get a number of additional A's equal to 2 to 
the power of the number we have already + one.  
[8:47] So our sequence is each next term in the sequence 
is the previous term plus 2 to the power of the  
[8:54] previous term plus one. And we should do that 
well through all 32 of the new bees that showed  
[9:02] up. And so the sequence goes five 69 that a lot 
and then a lot a lot a lot a lot. You get the  
[9:13] idea. Now to deal with the elephantized number in 
the room. What does a 22digit number even mean?  
[9:20] And a lot of people when I said a big number 
in a game might be thinking about the classic  
[9:24] 52 factorial because if you take a deck of 52 
cards, you give them a decent shuffle. There's 52  
[9:30] factorial possible arrangements of this deck. But 
here's the thing about that number and the card  
[9:36] number. We could write them down. I mean, I have 
written that one down. You could write down 52  
[9:43] factorial. But the next step up of this, could you 
write that down? because it's going to h well in  
[9:49] binary it'll have that many digits because that's 
the rate at which this is growing and in base 10  
[9:57] we just divide it by it's about a third going from 
base 2 to base 10 so it would have a third of that  
[10:03] many digits in base 10 the next one up and there's 
no way we can write that down the one after that  
[10:09] absolutely not the number of atoms in the knowing 
universe we believe is about this many Now, um,  
[10:18] I've made that number up. That's 81 random digits. 
It's about 10 to the 80. So, it could be this.  
[10:25] The exact number it happens to be right now will 
look exactly like this. And anything above that,  
[10:32] there aren't enough atoms in the universe, one 
per digit, to write them down. And this very  
[10:38] quickly exceeds a power bigger than the number of 
atoms in the universe. So, the order of magnitude  
[10:45] is just nuts. In fact, if you put this whole 
recursion thing, I mean, we tried it in Excel,  
[10:50] that crashes real fast. We put it into Python, 
that's as far as it gets. We then use something  
[10:56] called hypercal. And if you do continue the number 
of iterations, you can represent the number as  
[11:01] this, a power tower that's 30 high. So, what we 
have here, a power tower. You start at the top,  
[11:08] you work your way down. At the top, we have 10 ^ 
3.6 * 10 20. And we could write all those digits  
[11:13] out. There's a lot. um if we could even work them 
out. And then the next one down. So you calculate  
[11:19] each layer as you go down. There are 30 of these. 
As you go down each one, the next one has as many  
[11:25] digits as the value of the one above. And it's 
nuts how fast you you run out of universe to  
[11:34] write this in. Like this is just an insanely 
large but finite and very specific number. So,  
[11:43] what's amazing about playing this in Magic the 
Gathering is that you are technically playing  
[11:49] an exact number of Astral Dragons. And I reckon 
if you got that many, I mean, you might win this  
[11:55] game. And you can argue it's not infinitely large. 
It's finite. And it's an exact an exact value.  
[12:03] Now, this is a lot smaller than Graham's number. 
Graham's number is just it's like iterative power  
[12:12] towers or power towers although it was just 
a threshold. It was just like an upper bound  
[12:17] below which the value to the problem must exist. 
And even though this is vanishingly smaller for  
[12:22] some definition of vanishing and smaller, it's an 
exact value that exactly solves this problem. So,  
[12:28] in that regard, this could arguably be the biggest 
number that's ever actually been used by humans in  
[12:36] this game. It just so happens that it's so big, 
we're unable to compute it. Oh, wait a minute.  
[12:43] You remember rule 107.2. If you can't calculate a 
number, it equals zero. You can't calculate this  
[12:52] number. It exists. It's it's exists as much as 
the number seven does. And compared to infinity,  
[12:59] it's about as big as seven. But as humans, we 
can't compute that. So if you did ever play  
[13:05] the shenanigans in an actual game, and there's 
only three cards, easy to achieve. We found them  
[13:10] all. Like some of them are a little obscure, but 
these are all absolutely attainable cards. If you  
[13:15] somehow contrived to play this in a game, depends 
if if the adjudicators will let you get away with  
[13:23] this. If they insist you need to be able to 
calculate the integer value other than just  
[13:29] proving the integer exists, you've now got zero 
dragons. And that's the smallest possible number  
[13:37] of dragons. Now, if you've enjoyed following 
along with Oh, this ridiculous. My goodness,  
[13:43] you might be the perfect person for what the 
sponsor of this video, Jane Street, has in store  
[13:49] for us. Because speaking of shuffling things, 
they have a machine learning neural network  
[13:55] puzzle which is going to be explained by exploded 
matt. If you're not familiar with Jane Street,  
[14:02] they are a research-driven trading firm where 
curious people work on deep problems and they  
[14:07] believe that deep learning is the future of 
quantitative trading. and they have an in-house  
[14:14] machine learning team who work on neural network 
models which help drive their trading strategies.  
[14:21] They also have to build the infrastructure 
required to make those inferences and continue  
[14:27] that research. If you're at all interested in this 
kind of neural network research, Jane Street's  
[14:32] machine learning team have put together a neural 
network puzzle. They trained a neural network with  
[14:37] 96 layers and then they shuffled them and you 
got to put them back in order. And you know,  
[14:43] forget 52 factorial. To my naive understanding, 
this is like 96 factorial. That's not a hint. I  
[14:50] mean, I had a look at it. I couldn't solve this. 
But some people watching the video probably can.  
[14:55] If you do think you've solved it, please send 
your answer into Jane Street. I would love to  
[15:00] see how many viewers of Standup Math videos can 
solve this neural network challenge. Anyone can  
[15:07] do the puzzle. You don't have to care at all about 
a career in finance. But if you are curious about  
[15:12] Jane Street and the work they do with machine 
learning, I will have a link below and there's  
[15:17] like a QR code somewhere on the screen. You can 
check out their open roles and available programs.  
[15:23] That's the video. Thank you so much for watching. 
I really appreciate it. I also want to thank  
[15:27] Tabitha Grove who designed these phenomenal cards. 
And if you support me on Patreon, we're going to  
[15:32] put some extra content up on there about how 
Tabitha designed uh these fantastical artworks.  
[15:38] And uh big thanks to Matthew Franklin who is the 
viewer who wrote in and mentioned this ridiculous  
[15:45] thing in Magic the Gathering. And if you know 
any other ridiculous maths things in games,  
[15:51] gathering related or otherwise, please do let 
me know. And um yeah, we'll also link to the  
[15:58] original Reddit post and whatnot um below. 
Uh that's it. That's the whole video. Um if  
[16:04] you're wondering why the table's shaking slightly, 
Skyabad the dog has come through. Skye, over here,  
[16:09] over here. Over here, and around. You got to say 
goodbye. Over Sky, here. And up. And goodbye.