How many chess games are possible?

[tromp]

> For the chess problem we propose the estimate number_of_typical_games ~ typical_number_of_options_per_movetypical_number_of_moves_per_game. This equation is subjective, in that it isn’t yet justified beyond our opinion that it might be a good estimate.

This applies to most if not all games. In our paper "A googolplex of Go games" [1], we write

"Estimates on the number of ‘practical’ n × n games take the form b^l where b and l are estimates on the number of choices per turn (branching factor) and game length, respectively. A reasonable and minimally-arbitrary upper bound sets b = l = n^2, while for a lower bound, values of b = n and l = (2/3)n^2 seem both reasonable and not too arbitrary. This gives us bounds for the ill-defined number P19 of ‘practical’ 19x19 games of 10^306 < P19 < 10^924 Wikipedia’s page on Game complexity[5] combines a somewhat high estimate of b = 250 with an unreasonably low estime of l = 150 to arrive at a not unreasonable 10^360 games."

> Our final estimate was that it is plausible that there are on the order of 10^151 possible short games of chess.

I'm curious how many arbitrary length games are possible. Of course the length is limited to 17697 plies [3] due to Fide's 75-move rule. But constructing a huge class of games in which every one is probably legal remains a large challenge; much larger than in Go where move legality is much easier to determine.

The main result of our paper is on arbitrarily long Go games, of which we prove there are over 10^10^100.

[1] https://matthieuw.github.io/go-games-number/AGoogolplexOfGoG...

[2] https://en.wikipedia.org/wiki/Game_complexity#Complexities_o...

[3] https://tom7.org/chess/longest.pdf

[jmount]

Nice stuff, thanks for sharing that.

I remember from a lot of combinatorial problems (like cutting up space with hyper-planes or calculating VC dimension) that one sees what looks like exponential growth until you have a number of items equal to the effective dimension of the system and then things start to look polynomial.

BTW: I was going through some of your lambda calculus write-ups a while ago. Really great stuff that I very much enjoyed.

[qsort]

I wonder if/how that interacts with the new draw rule. (For the uninitiated: the formal rule to adjudicate games as draws automatically or on time is that the game is a draw if there exists no sequence of moves that could lead to checkmate. Interestingly, although this has almost no strategic implications, it means that... it's almost impossible to write a program to detect draws that's technically correct. A similar corner case is draws in Magic the Gathering, which is literally undecidable in general.)

[Someone]

I do not see how that’s a good estimate. For example, take a game length of, on average, 4 and a branching factor of 10. That gives an estimate of 10,000.

Chances are there are games of lengths 3 and 5, too. With that branching factor, there are 1,000, respectively 100,000 of those, for a total of 111,000. That’s over ten times as many games as estimated.

The larger the spread in game length towards games that are larger than average, the more the proposed estimate underestimates the actual number.

[GMoromisato]

One thing I always wondered is how many moves, on average, do you have to play before reaching a position that has never before seen on Earth?

Or maybe the question should be what percent of games reach a position that has never before been seen?

[recursivecaveat]

Apparently ~75% of the positions in the lichess database (as of 6 years ago) have only been seen once ever. Average game length is 30-40 moves, so for the completely average player it would be like 10+ moves I suppose. The stronger the players the longer it will take: I found some comments suggesting 20+ for high level players.

[tromp]

I think that the average chess game played between humans contributes between 20 and 40 new positions (note that a 30 move chess games has 60 plies).

[bdamm]

You'd probably need to make a determination of the skill of the players. A very strong player vs a novice could be scholar's mate most of the time.

[jonas_kgomo]

I watched a movie a few days ago and they basically said there are more states in the game of chess than atoms in the universe? https://www.youtube.com/watch?v=xfMQ7hzyFW4

[adonovan]

Sure, but in combinatorics the number of atoms in the universe (say 1e80) is not a large number. For example, the factorial of 59 is larger. If you own 30 pairs of shoes, there are factorial(60) ways to arrange the individual shoes in a sequence.

[paulpauper]

meh. I think it would have been more interesting had the author discussed more granular estimates. Mathematicians have narrowed it down more by considering the properties of the pieces and bijections.

[LegionMammal978]

Assuming you're referring to [0], that's a statistical estimate of valid chess positions (based on clever methods of uniform position sampling + fast validity testing), not valid chess games (based on estimating branching factors for very long games).

[0] https://github.com/tromp/ChessPositionRanking

[RA_Fisher]

Infinite. :) Chess is strictly unbounded.

[erehweb]

This link https://wismuth.com/chess/longest-game.html from the article talks about the 2014 changes (75-move rule and draw by 5-fold repetition) that make it no longer infinite.

[Mordisquitos]

That was also my intuition. Unless there's a rule that can stop two immortal but dumb-as-bricks players from indefinitely cycling through the same non-capturing moves surely the answer is 'infinity'.