# What is the average length of a game of chess with only random moves?

The famous Shannon number, 10^120, an estimated lower bound on the number of possible chess games, is based on the assumption that a typical game lasts 40 moves.

In games between skilled human players, this average game length seems to be about right (see this question), but if we want to count all possible chess games, then we also have to include the vast majority of games where the moves don't have any purpose or sense whatsoever. Such a game with random moves would surely be likely to last more than 40 moves?

So what is the average game length of a random game? This should be easy enough to simulate on a computer.

• – GloriaVictis Sep 9 '15 at 20:01
• @GloriaVictis I find it interesting that so many random games end in a draw. Almost all games with length above average end in a draw. – Dag Oskar Madsen Sep 10 '15 at 12:41
• @DagOskarMadsen: that's quite expected, as a lot of special things have to happen for a random position to be checkmate (check and no legal moves). It's much more likely to just run into the 50 moves rule at some point. – RemcoGerlich Jul 25 '17 at 7:42

For a twenty-five thousand games the average length was 357 ply.

An interesting question with which I've done significant exploration. My program Symbolic can generate random games using the /dev/urandom pseudorandom generator to any length. For a recent run of ten million random game, the average length was 342.064 ply.

``````[] rg 10000000
Checkmate  1527544 0.152754
FiftyMoves  2241451 0.224145
Insufficient  5358614 0.535861
Repetition   258221 0.0258221
Stalemate   614170 0.061417
Average ply length: 342.064
Maximum ply length: 974
``````
• With the large numbers involved, this is probably significantly smaller than BlindKungFuMaster's answer. So there must be a difference between the scripts? – Dag Oskar Madsen Jan 11 '16 at 10:15
• Symbolic probably claims a draw whenever possible, which to my mind doesn't make total sense. The script I reran probably never claims 3-fold rep (but I didn't check that). I think 3-fold rep or 50 moves should only be claimed by the weaker side (heuristically every second time) or not at all. – BlindKungFuMaster Jan 11 '16 at 12:28

One estimate of the length of a game of chess with all random moves might be 90 moves. This is the length of an average game as used by Shannon (40 moves), with 50 more moves added until the 50-move rule takes effect (at which point the game can be drawn if one player requests it). The first 40 moves will certainly simplify the board, as pawns can only move forward, and must start to capture other pawns and pieces. There may also be some promotions of pawns into queens or other pieces, and the pieces in general will be "thinned-out" from random captures. After this, pieces will continue to move, and still with no purpose. After 50 moves of no captures and no pawn promotions, either side can claim a draw. If we assume that a random player might also request a draw at a random moment (once being allowed to do so), then such a request is inevitable, and the game will end.

• Do the "players" also resign at random? – Philip Roe Jul 23 '17 at 22:31
• I have no idea if that's what the OP wanted as an assumption. If that counts as one of the possible "legal" moves, and there's an average of 30 moves at each ply, then maybe there's a 1/31 chance of a player resigning at each turn. – tomoka kazuki Jul 24 '17 at 4:37