How Complex is Chess Really?

It is claimed that the number of atoms in the observable universe is less than the number of possible games of chess.

The number of atoms in the universe is approximately 10⁸⁰.

This is 80 tens multiplied by each other.

 So, how is it possible that the number of possible games of chess is greater than 10⁸⁰?

Let us do the Math!

First, a chess move, by definition, is equal to an initiation by White and a reply by Black. So, “two moves” means that each player plays twice. This is a total of four plays.

Claude Shannon, a mathematician and a theorist estimated that each game has an average of 40 moves, where each player has 30 possible actions to take every time he or she wants to make a play on average. This makes the total possibilities of chess games 10¹²⁰ possible games.

How so?

Suppose it is move 1, player White has an average of 30 possible initiations and player Black has an average of 30 possible replies. So, the total number of possibilities for play 1 is (30*30) or 30²

Suppose it is move 2, player White has an average of 30 possible initiations and player Black has an average of 30 possible replies. So, the total number of possibilities for play 2 is also (30*30)

So now, we have a total of (30*30)*(30*30) or 30⁴ total plays of the first and second moves.

Suppose it is move 3, player White has an average of 30 possible initiations and player Black has an average of 30 possible replies. So, the total number of possibilities with the first, second, and third plays is (30*30)* (30*30)* (30*30)

So now, we have a total of 30²*30⁴=30⁶

And so on…

Look at the pattern

Move 1 yields 30² total plays by both players.

Move 2 yields 30⁴ total plays by both players,

Move 3 yields 30⁶ total plays by both players.

Move 4 yields 30⁸ total plays by both players.

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Can you see the pattern?

Shannon, after looking at a large number of games, estimated that the average number of moves in a chess game is 40. What is the total number of possible chess games? In other words, how many plays do we have after Move 40?

Move 40 yields 30⁸⁰ total number of possible different games.

Note that only after move 40 (40 initiations and 40 replies is a game going to finish).

30⁸⁰ can be rounded up to 10¹²⁰, which is greater than Shannon’s number of 10⁸⁰.

So, under Shannon’s condition of an average of 40 moves per game, where each player has 30 choices every time it is his or her turn to play, the total number of chess games is greater than the total number of atoms in the observable universe! 

See how complex chess is?