[HN Gopher] How many chess games are possible?
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       How many chess games are possible?
        
       Author : jmount
       Score  : 20 points
       Date   : 2026-01-27 19:59 UTC (3 hours ago)
        
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       | tromp wrote:
       | > For the chess problem we propose the estimate
       | number_of_typical_games ~ typical_number_of_options_per_movetypic
       | al_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 x 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 wrote:
         | 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 wrote:
         | 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.)
        
       | GMoromisato wrote:
       | 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?
        
         | tromp wrote:
         | 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 wrote:
         | 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.
        
           | reassess_blind wrote:
           | Yes, the stronger the players, the more often they will both
           | go deeper into established theoretical lines that have been
           | played before.
        
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