# How many chess games are possible?

How many games of chess are there? So, there is a popular fact that the number of games of chess is greater than the number of atoms there are in the observable universe. This is the fact. Is it true? Well, the number of games of chess is known as Shannon’s number so let’s find out what is Shannon’s number, and how it was worked out. Claude Shannon. He… in the 1950s he wrote a paper “How to program a computer to play chess” …It was about how to program a computer to play chess. And in that paper he came up with an estimate for how many games of chess there are So it is only an estimate. So he estimated that the number of games of chess would be Ten to the power…round about ten to the power 120 Which is… well, just massive! It’s billions and trillions of googles It’s so massive, that’s a huge number. If we compare that with atoms in the observable universe there’s about ten to the 80 atoms in the observable universe so there are more games of chess You could assign billions of games of chess for each atom in the universe. How did he come up with this massive figure? So what he decided was, well, he looked at some games of chess and he said Well, on average, at any position, there are about thirty legal moves that you can make. So, the first player would have thirty legal moves he could make And if you do two moves, that’s the first player then the second player Then that would be, for each move by the first player the second player would have thirty. so if was only two moves It’d be thirty by thirty So that would be NINE HUNDRED already, just with two moves Some terminology here: when I’m saying *moves* I actually mean what is called a *ply* That means that one player goes first, then the second player goes next, then the next player… So in chess terminology a move is the white player then the black player So another move would be the white player again, the black player again. But we’re just going to say *moves* to be: each player takes a move It’s actually called a ply in chess terminology. So if the first player has 30 legal moves he can make Then for each move the second player has another 30 legal moves he can make So with just two moves there are nine hundred moves you can make altogether. If it was three moves, it would be 30 by 30 by 30 If it was four moves, it would be 30 by 30 by 30 by 30. Shannon said, Well, a game is about 40 moves. Ah, that was in chess terminology, so he means 80 plays. So… This is all Shannon did, he said A game is about 30 legal moves… 80 plays and that is around about 10 to the 120. BRADY: Well that is the most wishy-washy… JAMES: Yeah. It was only in passing, it was only an estimate, it was only a paragraph of the paper And he did this rough estimate, this rough number, to show that if you had a computer and it was trying to work out then the future of the game, and trying to work out all the legal moves and where this game was going to go so that it could make decisions, sort of, how to play next… Then, the computer would never make a move If it was calculating one game per microsecond it would be until the end of the universe it would never play. This was the point Shannon was trying to make with this rough estimate. So we’re going to have a look at a sequence of games there are when each player takes their turn. So the first move is by white. So…let’s take the first move, and white has twenty legal moves. So we’ve got…we’ve got one here for each pawn, so that’s eight. We’ve got the double move there by each pawn, so we’ve got sixteen. We’ve got two moves there for each knight…two moves on this side here and here… And that’s twenty. So, twenty moves for the first player. Now black takes the next turn And for the next turn he can respond with the same twenty moves. So for each move player 1 takes, black can take twenty moves, so you multiply, it’s 20 by 20… There are now 400 games that could have happened, already. With just two moves there are already 400 games that could have happened. Let’s see what happens next. Then suddenly it becomes much more complicated. When white plays next it becomes…8902 moves. Suddenly, phoof! 8000, well, nearly 9000 games that you can play, with just 3 plays. BRADY: This sounds a lot more precise than what Shannon was doing, this is real numbers. JAMES: Yeah, these are real numbers. With these small numbers we can work this out. So far so good — it’s going to get harder with larger numbers. So, let’s write out a few more. The next move is black, and he’ll have 197,742. So after black moves again we now have 197,742 possible games that could have been played. All the possible games we could have done in those four moves is already 200,000. BRADY: This is crazy. JAMES: Huge numbers already, huge numbers. BRADY: Although, you are able to know exactly what the number is, so it just seems like if you spend enough time on this you’ll be able to… JAMES: So OK, we can keep going can’t we? So we can keep going and these numbers are going to get bigger and bigger Now, in theory, the total number, the largest number, the longest chess game can be… something around 11,800 of these plys. 11,800. That’s invoking though, you have to invoke the 50 move rule, cause you could just go on forever, if you just ended up with two kings, just going backwards and forths. A game could last forever. So there is a rule that stops a game lasting forever that says: Well, if you’ve played 50 moves with nothing being captured, and you’re just messing about– BRADY: Or a pawn being moved, isn’t it. JAMES: Yeah it’s a pawn hasn’t moved, something hasn’t been captured, or you repeat the board three times. If that happens then you call it a draw. So there’s a cut-off point. Now some people have worked out what the longest possible chess game is… It’s something around 11,800. There’s some disagreement, not much disagreement though, Within, a hundred or so of that. So if you see how fast this sequence is growing, can you imagine how many games there are altogether. Especially when you’re going all the way up to nearly 12,000 moves. Some of those games would be nonsense games, where you can win, you’ve only got one move left and you can win in one move, but you don’t, you start moving other pieces and you start doing other things. It becomes a massively complex tree of possibilities and that’s why this number is, just, going off to a HUGE numbers. Godfrey Hardy, a famous 20th century mathematician He tried to estimate the number of chess games there were. His estimate was ten to the power ten to the power fifty. Let me say that again: it was 10^(10^50). This is miniscule when you compare it with what Hardy’s estimate was. What Shannon was doing was saying, “This is a forty move game, where the average number of moves is thirty.” So he was saying that if there were 80 plays, that’s what he’s saying, He’s saying this is about ten to the power 120. That’s what Shannon is saying. So he’s not even considering all these other games. So it was Godfrey Hardy, the famous 20th century mathematician – worked at Cambridge, discovered Ramanujan, He tried to estimate how many games of chess there were. It was actually when he was writing about Ramanujan, cause Ramanujan had sent him a paper which had a large number in it He said, “Just to understand how big this number is, if you compare it to the number of games of chess I reckon that’s 10^(10^50).” Was Hardy close? I don’t know. He didn’t give any working out for this, this was in passing I did say a lot of these would be nonsense games; let’s try some sensible estimate. If it was, let’s say if each player had an average of three sensible moves, instead of thirty legal moves. Same sort of idea. If we did that… So instead of 30^80 it would be, say, 3^80. Does that seem more reasonable? Yeah? So that’s 3^80, I can tell you that’s around about 10^40. So now not as large as the number of atoms in the observable universe Still, still very large though, 10^40. If for example, everyone in the world paired off and they had to play a game of chess every day and it had to be a different game every day, and they did that… To play all possible games, this 10^40 sensible games it would still take you trillions and trillions of years to play them all. Or if you think about it another way If we consider all games of chess that have ever been played in history, then that is only a tiny fraction of all the possible games of chess there are to play. BRADY: I can’t think of a better supporter for a video about chess than a company called squarespace. Now if you want a presence on the web, whether it’s a blog, a business, or just a cool way to tell the world about what you’re doing, then squarespace is a super resource. I host my own blog on squarespace, and it’s so quick and hassle-free. In a matter of minutes I can have anything up on the web, looking great — and that’s looking great on both computers and mobile devices, which is often the hard bit. Now whether you’re a bit of a novice, or you are a coding grandmaster squarespace will give you as little or as much control as you like. So whatever your endgame, you really should…check them out (see what I did there? “check them out”). “endgame”, it’s pretty good that. Anyway. They’ve got a great range of templates to get started. Then you can tweak them, play around until your heart’s content. And I’m not just saying this because they’ve sponsored today’s episode which they have, and thank you for that, but I am a happy customer. Now you can have a 14 day free trial, give it a go before you commit, no credit card required, and then if you do like them, you can get 10% off your first purchase. Go to squarespace.com/numberphile And thanks to squarespace for supporting this episode. As they like to say, “Build it Beautiful” BRADY: Do you play chess? JAMES: I do play chess, I know chess, I’m no expert on chess, I’m no grandmaster. I play chess for fun perhaps. I used to play chess with my dad.

dont get mistaken with the number of how much different positions are possible because that number is much lower

Well technically if somebody didn't call stalemate that game would go on forever

as far as i know the number of possible game variations is in SKAT even higher!

Theres gotta be way more than 10^80 atoms in the universe

Players have 20 legals first moves not 30 (8 pawns x 1 OR 2 squares + Knights have 4 options) why not use 20 (the right #) for ur example instead of 30. I mean thats just irritating

movesSneeky Chess reference 'in passing'.

This is such horseshit…

how was the 3 and 4th moves calculated

of course this depends on the previous move.

8902/400 is 22,255

but what i don't get is if white moved a knight that the number of moves would go up to 21

so how was the number 8902 reached?

Actualy you can play for endless period of time :/

Im guessing that their estimates are wrong because they dont the en passont rule

how 8902 in the third move is calculated?

I don't know if anybody noticed but checking the comments no one mentioned that there are only 20 legal moves for the first turn, thus only 400 combinations for the first turn alone (not 30 x 30 = 900).

Your other videos are great but this one is quite frustrating actually.

Whoa my mind just exploded

The number of how many different setups 7^64

This is just a joke tho

64*32 ez

4 moves = checkmate

"There is a popular fact that the number of games of chess is greater than all the atoms in the obersvable universe. But is this true"

Since ypu failedto answer this question. I will.

No, that is obsurd. 7*10^27atoms in the HUMAN BODY! The number of atoms in the observable universe DWARFS the number of games of chess out there.

All we now need is a learning matchbox computer for chess

math + chess = perfect

Dude built like a dorito

30^80≈10^118

what are supposed to do with this information?!

But google's alpha zero mastered chess in just 4 hours and defeated stock fish 8 , the most powerful chess engine.

If the point is to say how many possible games you need to calculate the actual amount, no matter whether it was logical

Can’t they both just play a Bishop going to a square that’s not check, and then move it back and do that an infinite amount of times

Ok sorry I just watched 6:40 never mind. Also I remembered the threefold repetition rule anyway

Kasparov has played them all

wouldnt you have to take into account moves that resulted in the game ending?

In the first second I noticed the guy is nerd enough so I decided to keep on watching.

O wait you can resign at any time. Take that into calculate

I love the idea that the numbers of possible chess games are finite.

I just clicked on this video because I didnt understand the Question in the title, lol

I never understood this question, there are actually infinite games, you can repeat the movements infinitely, changing once every 3 to avoid draw.

I commented before ending the video.

It is infinite really

The guy give me creep

-1/12

Technically there is an infinite number of games that could be played as one player could just move a piece such as the queen back and forth an infinite amount of times adding one of for every game

Is it one of np problems

3:11 Can someone call a priest? No, seriously

Thank you. Now I know that i am really just bad at chess, i always thought that the computer knew what play i will do. I feel even worse now.

So many possibilities yet everyone plays the London

Hypothetical possible number is literally infinite. You can cycle the same series of movements over and over for both sides with slight variations each time so they do not repeat. Assuming you do not put the 50 move rule in.

There is an infinite amount of chess games possible…

More accurate to ask how many legal positions are possible, as many exact endgames, with exact pieces on squares and other board considerations can be reached multiple ways

Can't wait to get quantum computers so we can finally calculate the true value of this sucker

This video is 10 mins too long

number of 52-card deck shuffles is ~ 10^68

Moral – Never play a chess game.

technically wouldn’t it be infinity since at the end you could have your king run around the board and your opponent will be in on it as well therefore your game will never end and every move will create a different way you can play

There is an infinite amount of chess games if we think about that the 2 players can for example move the same queen for example back and forth without doing anything, and that can be done not with the queen only, so there is an infinite amount of chess games.

at 3:45 cant he do the rook king thing…

Isn't answer simply ∞, because you could in theory just make buttons not to eat each other?

10^120 isn't even that big of a number

I lost it at 10:01

10¹²⁰

wait what if a game was rigged so that each player moved there bishops backwards and forwards

There is only one ideal played chess game.

30 legal moves for the first move is wrong. There are 2*8 moves for pawns and 2*2 moves for Knights. No other pieces can start the game. So there are 400 variable lines for 1. … …

10^120 = 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Actually how many possibilities you have for your second move will depend on what your first move was. If you first moved the pawn in front of a bishop, that will give you fewer chances for the second move, since the bishop (unlike the other figures) can't move ahead to the now empty square, and the knight can't move to the position where you placed your pawn, among other factors conditioning how many moves you have.

My point is that for your first move you've got twenty choices, but how many choices you have for your second move is not a fixed number. And this variability gets larger and larger as you keep playing. So I wonder how you can remotely calculate how many possible different moves a whole game can have.

i would like to watch a 11k move game 😀

this doesn't answer the question at all because games have a goal. people act to achieve the goal. they don't make nonsense moves!

10 to the 120 ways im a loser.

There are more than 10^40 sensible chess games. There are more than 3 sensible moves

I don't know have I haven't known this for this much time but we human just really love to make games that have more move than that thing like atoms or infinity

How many chess games are possible?

a lot

someone else feel the grima from that sound of marker writing on paper? i like everything about this channel besides that…

nice bro, in the future, i will create a computer that can play all of those games. Wish me luck.

moooooove

Immortal game

"How many DIFFERENT chess games are possible?" is what you should've named it, but the video was awesome so it doesn't really matter.

It’s an infinite number. You can just infinitely do the same move over and over again

It’s an infinite number. You can just infinitely do the same move over and over again

You don't know how many particles there are so stop lying.

5:23

….I am a chess player….in the first move (two ply) there are 20 possibilities for each player…16 pawn moves and 2 moves for each night….therefore there are only 400 possibilities in the first move…

I hope at some point in time a super-duper-quantuum-computer can calculate the correct number xD

Yet like a 3rd of games end in 4 moves

Önli

This does not even count games like chess 960 with different starting positions.

Giri would draw them all

Ok, how about games of go?

What is the probably that someone ever played the exact same game of chess as someone else? Through all time?

I don't believe that there is a limit, especially if the 50-move rule in the end game is not applied. Or if the 3 repetition rule for draw is absent.

Let's go champ!

6:33 the 50 move rule is not about 2 kings left on the board. Since at that point it would already be stalemate.

I dont think this is correct, because it doesn't consider the spot that other figures occupy

6:33 Actually… no, according to FIDE (Fédération Internationale des Échecs, the biggest chess federation in the world), a player has to claim a draw. If neither player claims a draw, the game will simply go on.

Obviously I understand that this is a boring correction… xD

this is why you should play chess. you'll almost never end up with the exact same game

9:16 in passing EN PASSANT LOL

At 1:30 2 of those moves are illegal

müüüves

“If you had a computer trying to work out the the future of the game and all the legal moves and where the game would go, it would never make a move“

So that's why Windows 7 chess lvl 10 wouldn't ever do anything

Will quantum computers be able to solve this problem with exact numbers?

This is the most unlikeable guy I’ve ever seen. There’s nothing he’s done wrong but I just hate him, he’s the real life mr. Hyde

The London system players always play the same game

Hmmmm. something strange in this calculation. Shouldn't each number to devide to previous number clearly(without modulo)?

like 8902/400 and 197742/8902

At least 17

*throw chess in fire *

I am out

me:how many posabilities did you see

my friend:infinite

me:how many of em do i win

my friend:none