# Are chess games repeated

If chess games, in the history by all players, were played randomly, could we have repeated games?

Shannon number is a lower bound for all possible chess games which is $10^{120}$. Suppose that we already had $10^{15}$ chess games (I have no idea if this number is accurate) in the history, could they repeat?

Thanks

Edit: Let me add that we eliminate short games. In fact my question is: If you choose a number from a set of $10^{120}$ and repeat that for $10^{15}$ could you obtain repeated numbers?

• It is a safe bet that some of the very short games have been repeated throughout history, perhaps even once for every single beginner of chess. Commented Feb 7, 2018 at 22:34
• Your question seems poorly posed at present. You'll need to be very specific about how you choose to eliminate short games to remove any ambiguity. Note that the problem of selecting from $10^{120}$ elements for $10^{15}$ trials is very different than the problem of finding terminal chess games over some threshold of moves as we have no reason to believe that all possible chess games are equiprobable with random play. Commented Feb 7, 2018 at 23:12
• A chess game being played randomly could be very different from a chess game chosen uniformly at random from the space of all possible games. If one plays random moves then the short games will naturally be significantly more frequent than the long games. Commented Feb 7, 2018 at 23:34
• An immense fraction of those theoretically possible games is extremely unlikely to be ever played in a real game. Even a game between very weak players contains not only purely random moves. I would not be surprised , if there were duplicates of games of length , say , $40$ moves. I have no source about the record length in this category. Commented Jan 16 at 11:08
• @ErickWong If random moves are made , the game will almost surely take very long (if we assume that noone gives up or offers/accepts draws. That a mate is reached is extremely unlikely , so such games will extremely likely end in a 50-move repetition or a repetition of a position 3 times. Commented Jan 16 at 11:11

Yes, under these assumptions, there are very likely some repeated games. The reason is that there are some very short chess games. For instance, Fool's Mate ends in 4 half-moves. Each of those 4 half-moves has under 100 options, so the probability of any one game following that path is at least $10^{-8}$. With $10^{15}$ games, you are very likely to have had at least two such games.