# What is the average number of legal moves per turn?

For example in the first turn there are 20 possible legal moves (16 pawn moves and 4 knights moves). This number increases in the middlegame and then decreases in the Endgame.

Are there some calculations on the average? Google search seem to be focused only on "average number of moves per game". Would appreciate if somebody have some links to some scientific researches or also blog posts.

• If I understand your question correctly, you would have to average over the number of all possible chess positions, which is an extremely large number. Besides, I am not sure how meaningful this average is (might depend on what you need it for). Commented Dec 4, 2018 at 17:36

What you are looking for is called the branching factor, and I've always seen the number 35 mentioned, but I don't know what the original source is. I guess someone estimated it some 50 years ago by counting the number of moves in a number of random positions from games, and then it became "common knowledge". The number 35 is reasonable enough in practice, but of course is not exact.

In today's age of big data, it should be easy to take a game database and answer the question of what is the exact average number of moves per position in the database. I'm sure someone must have tried that already, but I haven't seen it.

• Correct. Not a very hard exercise but pointless. Commented Dec 5, 2018 at 4:11
• Are those 35 for a typical game of chess or an average over all possible positions of chess? A game database would not contain all positions, but only a fraction of possible positions which could affect the value. Commented Dec 5, 2018 at 15:15
• @user1583209 I think it is the average of the played games. Commented Dec 6, 2018 at 15:46

Whilst acknowledging the comment by @SmallChess that this is pointless, it is also relatively straightforward to do. I analysed 2,539,871 games from a ChessBase mega database counting the number of moves for the next player to move before each move was played. I did not include the number of moves available after the final move of the game had been played. The total number of moves played was 194,389,820 (76.5 per game) and the total number of available moves was 6,039,013,721 giving an average of about 31.1 per move.

Here is an analysis of the average number of moves available for each colour at each move point for the first 100 moves of each. Move 0 is prior to the first move of each colour. It is interesting to observe that, on average, black has fewer options than white. I am not sure why this would be-hopefully, it isn't an error in the analysis (!) but I don't think it is.

• +1; really nice effort! Any chance to also report some confidence intervals? Commented Apr 30, 2019 at 7:43
• @GloriaVictis I can try, although I haven't coded that before. For a 95% confidence interval I am getting a margin of error around 0.01 after about 30 ply and about 0.026 after 100 ply. Are those the sort of values you would expect? What form would this data be most useful in? Commented Apr 30, 2019 at 19:15
• it's not an error. I did the same analysis and arrived at the same result: on average, white has more available moves.
– Marc
Commented Aug 31, 2019 at 19:53
• I think that makes sense; Black is more likely to go into a defense such as the French that limits one or more of his pieces.
– D M
Commented Aug 31, 2019 at 23:31
• What distribution would this be, lognormal? What are the parameters for these distributions? Commented Mar 8, 2020 at 23:29

The variability in number of legal moves is about the same for black & white... except for that odd little bump for white on move 7. This phenomenon demands further research!

• Some very interesting data. Is that bump actually just on move 7, or is it spread out over moves 6 and 8 as the graph would suggest? Commented Aug 31, 2019 at 21:43
• The anomaly is indeed spread over moves 5, 6, and 7, with its peak at move 6. I'll add a bar chart that makes it more clear.
– Marc
Commented Sep 2, 2019 at 8:03
• Perhaps at this point in the game W is more likely to have castling opportunities that B doesn't? There's a smaller anomaly bump for B a few moves later. Commented Mar 7, 2023 at 21:39