# What is the number of chess position for each move? [closed]

I have read some articles that chess has a numbers of 10 to power of 120. Can some one can throw lights to it:

White 1st move: No of Position White 2nd move: No of Position till White 72nd move: No of Position

Similarly for Black

Black 1st move : No of Position Black 2nd move : No of Position Till Black 72nd move : No of Position

Your help will improve one's chess understanding better and will bring some hope to readers relating to chess can be solved.

You are misusing the word "probability" in your question. 10 to the power 120 is known as the Shannon number and is a lower bound for the game-tree complexity (number of different games of chess). This has nothing to do with probability. Also this number of different games is not the same as the number of possible chess positions (which is estimated as 10^43).

If you look at the starting position in chess, each player has 20 possible moves (16 with the pawns and 4 with the knights). So the number of possible games of 1 move (2 plies, i.e. one white move and one black move) would be 20*20 = 400 already.

Now the number of possible moves increases initially because your pieces get more space. The number of possible second moves is not fixed, but will depend on the first move. So it is impossible to give an exact answer to your second question. But you can do some estimates...

If you say that at each move you have a choice between 30 moves (this is an estimate) and if you say that an average chess game lasts 40 moves (80 plies), you end up with the number of possible games equal to: 30^80=(30^2)^40 which is approximately equal to 1000^40=10^120.

This is a very rough estimate but is sufficient to show that chess is incredibly complex and is not going to be solved anytime soon.

You might also like the Numberphile episode on the topic

• Where is the proof, that 10^120 is a lower bound to the number of possible chess games? This statement certainly does not follow from Shannons extremely rough estimate. – Simon Fromme Feb 4 '17 at 19:16
• @SimonFromme: No proof as such, but Shannon in his paper says that the number of around 30 possible moves is fairly constant throughout a game (empirical evidence from master games) and also says that the number of 40 moves for a game is rather conservative, since in this context one need to consider the number of moves until mate not until resignation. – user1583209 Feb 4 '17 at 20:38
• Well that is nothing that would justify saying "10^120 is a lower bound for the game-tree complexity". All this is is a rough semi-educated guess. – Simon Fromme Feb 4 '17 at 20:45