Has the number possible chess games ever been calculated?

hey guys look what i found youtube.com/watch?v=Km024eldY1A

Brute force is the only way to calc a solid answer. It'd be such a boring, worthless exercise; other than learning more programming skills, gaining experience.

Then again, we do know how to calculate number of permutations or combinations of elements in ordered sets without resorting to enumeration.

Infinite, technically.

56. Kg2 Ka6
57. Kg1 Kb6
58. Kg2 Ka6
59. Kg1 Kb6
60. Kg2 Ka6
61. Kg1 Kb6
...
...

good one m'lord, upvoted

Your sarcasm is retarded, the question was the amount of possible games. This is a possible game: Claiming a draw is NOT mandatory.

Even if you want to backpedal and say that claiming a draw is compulsory for this exercise, the amount of combinations is still arbitrarily large: You can keep repeating moves twice so that the requirements for a draw never arise but you're still repeating moves a lot.

Yeah, but if you go 50 moves without capturing a piecce or advancing a pawn, it's a draw.

It would be fun to calculate the longest possible chess game based on those rules.

Your post indicates you have no idea how Chess is played. You made a reddit tier attempt at being witty and you need to go back.

So what's stopping you from doing this?

56. Kg2 Ka6
>49 moves of nothing
106. Kg2 Ka6
>49 moves of nothing
156. Kg3 Kb7
>49 moves of nothing

You can abuse the 50 move rule and the 3 move repetition rule and create an arbitrarily long game by just doing 2 repetitions with 49 moves between them each time. Add in that you can promote pawns and the number of moves possible before a draw is compulsory is so arbitrarily big that even asking this question is pointless.

You are actually retarded. The question never had anything to do with how people play chess. Are you saying that a game in this exercise can't have dumb blunders all over the place? Fuck off with that.

Thing is that moves in chess are dependent on previous moves.
As is usually the case with permutations and probability, dependencies make things much harder.

Just having a king on a board and looking at the possible moves is a moderately difficult problem for which you need graph theory or random walk.
Now, if you have two kings that can't approach each other, the problem becomes very hard to solve by symmetries.
Adding even more pieces also with different rules pretty much guarantees that there is no easy solution.

My point was that there is a finite number of possible games. Eventually you run out of pieces to capture and pawns to push.

If there is a limit on how many times you can be in the exact same configuration, then games cannot be *arbitrarily* long. They could be pretty damn long but there's a constant upper bound on how long.

I'm thinking now that counting all possible games is probably harder than coming up with a winning strategy. For the former you have to consider situations that would never arise in reasonable intelligent game play just because they're possible.

Maybe so, but a low-ball for the amount of possible legal states of the board is 10^50. If you take that number and consider how many it would be if you're taking into account complete games, I think anyone would feel comfortable saying it's far beyond the number of atoms in the observable universe. Therefore, I stick by my philosophy that the question is meaningless and it's enough to just say that it's an obscenely high number.

I don't think anyone ever considered the exercise meaningful. A googolplex is a large number, yet still finite.

>So what's stopping you from doing this?

>56. Kg2 Ka6
>49 moves of nothing
>106. Kg2 Ka6
>49 moves of nothing
>156. Kg3 Kb7
>49 moves of nothing

The 50 move rule, you have to capture a piece or the game is drawn.

>my philosophy
you mean
>hurr infinite cuz you can go back and forth
?

fuck off to reddit honestly

what's with your fixation on reddit?

You're actually a retard, you don't even understand chess.

Your only worthwhile contribution to humanity will be your own suicide.

what's with your fixation on making 0 content, personal posts that don't say shit and only encourage people to call you a retard? post your face if you like attention this much, otherwise fuck off

Chess complexity according to wikipedia is EXPTIME-complete without the 50 moves draw rule and with 10^123 possible games

You're awfully cranky, I think you need a nap.

okay here's what you want
>YOU
>YOU
>YOU
>fuck off you retard

You know, I can't reply to you if you don't give me my (You)'s

>define what you mean by number of possible games
All unique legal variations between opening and conclusion.

The 50 move rule is optional, in that a player must say "I am beginning a 50 move count" and then start counting. If no player decides to initiate a 50 move count (and there are situations where a draw is as bad as a loss in tournament play, so this is a reasonably possible occurrence) then a game can continue indefinitely.

As such, chess is unsolvable in the mathematic sense, assuming permutations which result in the same position are considered different games.

You have to claim a 3 move repetition, too. I think the inherent assumption is that all draws are forced.

Enlighten us on how the question in the OP is worth answering. What do we gain from knowing the number of potential games? What do we gain from knowing anything more than it being an immense number? You understand that, as scientists, you shouldn't be wasting time on questions that take more effort than they're worth, right? If the answer to this helped us understand AI or the human mind more, it would be worth helping about. It doesn't, it doesn't help us figure out anything new at all.

>I disagree with you about chess, you should kill yourself
Esteem issues much? I know you're probably overcome daily with self-doubt about your own worth but there's no need to take it out on others.

>On an anime image hosting website
>Complaining about wasting time

Oh, the irony of this post!

Also, you do realize
>& Math
is part of this board name, right?
Mathematicians pride themselves on being useless.

64!

Nice counter-argument, the green really adds.

If you look at all the great accomplishments mathematicians did, they all had a purpose. Nobody is going to win the nobel prize for knowing how many possible chess games there are. It's not going to advance humanity in any way. Mathematicians don't PRIDE themselves in doing nothing, you do.

What the fuck, the definition of pure mathematics is literally math that doesn't have any purpose in real life

I don't disagree, theres nothing to disagree on, neither the rules of chess or your complete fucking ineptitude are up for debate.

Please kys

I don't know and I don't know how I could know. Chess boxing is a chess game and the number of possible right jabs that could be thrown in the first round is immense but maybe a finite number because of atoms but honestly like I said idk.

>ITT: people who don't actually know how to play math

you cant have infinitely long games you retards, its in the bloody rules

Yes you can, you can't repeat the same move over and over but you CAN repeat a cycle of moves over and over.

no you cant, the game ends if the same configuration is reached 3 times, and since the number of configurations are finite, no game can be infinitely long.

Wrong. You can claim a draw for threefold repetition, but you're not obliged to do so so infinitely long games are possible.

The game is supposed to end then, and does in in electronic games, the only reason for the 'You can claim a draw but you dont have to' part is that someone doesn't actively check if the conditions are met for the draw, so its on the people playing to notice it.

You're forgetting that every time you touch a piece, you shave a few moleules off it, so there comes a point where the pieces have been completely worn away to nothing.

Also it's a stalemate if the sun burns out during the game

This is a stalemate, you stupid fuck

I'm sure that's not in the rules.

You'll have to go over 100 years (probably much more) back in time to find anybody playing with this ruleset.

Modern chess (which I assume is the subject of this thread) forces a draw when any one (path-independent) configuration is reached three times, regardless of what any player claims.

/thread

theres about 10^120 board configurations
youtube.com/watch?v=GP9bhfi-SyY

I have an idea;
Consider all possible arrangements of pieces you could place on an 8x8 chess board. Obviously some of these arrangements are impossible to reach in the course of a standard game (like one side having 10 queens, or nine pawns). These impossible-to-reach positions are called "illegal positions" since no game of chess played with standard pieces and legal moves only will ever reach them.

The rest of the arrangements are called "legal positions". Some of these legal positions are actually checkmate positions. We may want to consider the set of all checkmate positions to be the set of all "complete" chess games.

Therefore, calculating all the possible checkmate positions will give an exhaustive list of all possible final moves, because from each checkmate position there is exactly one move that leads to that particular checkmate.

This list of legal checkmates is the lower bound for all possible games, and the upper bound is the minimum number of moves to play backwards from all checkmates to reach the starting position.

>because from each checkmate position there is exactly one move that leads to that particular checkmate.
I'm not sure this is actually true.

Okay well again, you're being a little obtuse here. For the sake of the problem, assume that the game ends after threefold repetition like it should and disregard human error.

>because from each checkmate position there is exactly one move that leads to that particular checkmate.
Just at a glance my intuition tells me that this is demonstrably false. After 10 seconds of thought I am sure that it's demonstrably false.

let me save you some time trying to demonstrate it and tell you why you're right and why i was wrong

consider the fact that in a checkmate position, there is a piece that previously moved, checking the losing king. That piece, prior to delivering checkmate, has to be in a legal position, and then has to make a legal move that results in checkmate.

You're right; there is no singular pre-checkmate position. But depending on which piece is delivering checkmate and where the checkmate occurs, there is a finite set of possible moves. If a pawn move delivers checkmate, there are three positions that could have resulted in checkmate (one where the pawn advances to an empty square to deliver checkmate, and one where the pawn makes a capture from either diagonal to deliver checkmate. For a bishop or rook there are up to 15 positions, and for a queen there are up to 28.

In a situation where checkmate results from a double-check, the piece that moves is the one that we consider to be the one that actually is delivering checkmate.

tldr; there is no one move that delivers checkmate but there is a finite set of moves that delivers checkmate, which is good enough

>For a bishop or rook there are up to 15 positions
should be 14
also I forgot that a knight could deliver checkmate from up to 8 possible positions

en.wikipedia.org/wiki/Shannon_number

It's probably infinite

10^120

That's also the cosmological constant.

Coincidence?

It's probably not infinite.

it's an asspull number though

>all these people misunderstanding chess rules
A naiive way to calculate the maximum number of moves is by counting the number of moves that delay the 50 move rule: 16 pawns * 4 moves (let's assume that average, some need to be captured for others to promote) plus 16 pieces captures + 8 pawn captures, total of 88 moves, times 50 moves is 4400 moves. We can ignore the three times repetition. Thus, chess games can't possibly be of infinite length.

Does that have three times repetition? Because it seems pretty strange to not include 50 move rule.

Claiming a draw is not mandatory, but I'm not a FIDE arbiter. The arbiter has the power to force it though. There has been a case where a nine move repetition occurred before the arbiter intervened because the guys were fags.

I'd like to see a fide rule that says three move repetitions are a mandatory draw.

One of you must be right.

Well it's finite because the pieces and board are finite. It would boil down to what actually defines a "game", more plausible is calculating the number of positions possible. The number would be so enormous as to be almost meaningless and incomprehensible.

How can a fixed, finite game be EXPTIME-complete? Are they talking about some version of chess that grows, like on an n by n board instead of 8 by 8?

Because normal, standard chess should take constant time. Just a really, really fucking big constant.

The "50 move rule" and the rule saying that it's a stalemate if you hit the same configuration 3 times, these are the only things limiting the length of the game.

Without those rules not only would there be an infinite number of possible games, the infinitude of games would be uncountable actually

An easier but still interesting question to answer might be the practical longest playable game.

Construct a graph where each valid game board is a node and an edge between nodes shows that a legal move links those two gamestates.

the result is a hybrid graph, a partially directed cyclic graph, although there will be huge regions of the graph where sub-parts are essential DAGs. What you have then is a longest path problem (NP-complete), which you will have to bruteforce the solution to.

You might be able to come up with approximations. If you limit the number of permitted cycles so you don't end up with technically infinite path lengths and you just try to connect sub-regions of DAGs to each other, and then brute forced it, you might be able to come up with some estimate of the longest game.

You can avoid cycles by defining a "gamestate" to include the entire history of the game rather than merely the current positions of the pieces.

By the 50 move rule there is a finite number of possible gamestates (including history)

Yeah, but when you get to 10^40 times more x than there are atoms in the universe, you're really talking about fantasy numbers.

never stop, you are my hero

It's heavily implied.

Would it really? It's a meaninglessly high number, one most people can't even name or read out loud.

I'm sure it could be managably expressed in googolplexes (googolplices?)

nah son
we're just talking about how many atoms there are in 10^40 universes

I can't believe it took like 30 posts to get this response...

chess is a very simple game compared to go

yeah but it's easier to mathematically solve a question like "what are the total number of possible go games"? because the rules are much more mathematically simple

I don't agree with that at all. The bound on the number of possible 200 move go games is somewhere less than
3.2×10^511.

That is not a very tight bound at all...like 300 orders of magnitude higher than a chess game of any length.

"the answer is a bigger number" isn't the same as "it's harder to find the answer". Nobody bothers with Go because with some rule sets it's not as interesting

More like 10 Also, it's not a very good or interesting response.

>the number is bigger
>therefore it's more difficult math
kek

>in that a player must say "I am beginning a 50 move count"
I thought that sounds retarded and looks like you're wrong
>en.wikipedia.org/wiki/Fifty-move_rule
You only have to claim the draw after the 50 moves have already happened.

Well, you gotta have at least 1 capture every 50 moves.
So take A_n the sequence of the number of pieces on board at round n. It's lower-bounded by 2 (both kings can't be taken). Assume you got a play wity infinite moves, then A_(50*n) is a sequence of integer stricly decreasing ; bam.

There's no games with infinite move, and an upper bound to the number of moves is 50*30 (30 is the number of takable pieces)

Now, all your games has a finite number of moves, and finite number of possibility for each move, bounded by 16*64 (max number of pieces * positions).

Make that an upper bound of 2^(1500) games. This is by far over-estimated tho ; but still finite (even for Wittentard).

>Well, you gotta have at least 1 capture every 50 moves.
a capture OR a pawn advancement