How many possible games are there?

How many possible games are there?


How many different chess games is possible to play? 

At the beginning of a game, the white ones chose one between 20 possible moves. Next, the black ones had 20 options of answer, any one that was the movement of the white ones. So, in a chess game, after the first movement it can arise 400 different positions. After 2 moves the number of the possible positions grows up till about 20.000. After few moves, the figure is enormous.

Claude Shannon worked out the total possibilities of the chess games

10120 (1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000, 000,000,000,000,000,000,000).

Shannon number (1) is an huge number, even larger than a googol (2) that the name of a known search engine.
In there 10120 possible games, there are a lot very bad, and there are a little percentage of good games. For instance, we can imagine that for each thousand million of possible games, only one of them is a good game. This would say that it would be 10111  possible good games, so, the potential of  chess beauty  is inexhaustible and that remain an enormous number of very good games that never have been played, the majority of this will never be played.

Claude Shannon                 Carl Sagan explaining the googol in the serie Cosmos 


(1) Shannon number
The Shannon number, 10120 is n estimation of the complex tree of chess games. It was for first time worked out by Claude Shannon, the father of the information theory. According to his results, it happen average 40 moves in one chess game, while each player choose one unique move between 30 possible ones .(In fact, it could be zero possibilities as in the event of checkmate or stalemate,  or so much as 218). So, there are (30×30)40 or 90040 different chess games. Approximately it is said that this number is the same as 10120 as a result of the equation:

90040 = 10x, then  x = 40 × log900

Nowadays, the complex of the possible tree games is worked out about 10123 (The number of the legal positions in a chess game is between 1043 and 1050. As a comparison the atoms number that exist in the universe is between 4×1078 to 6×1079.


(2) The Googol
A googol is large number, equivalent to 10100  (So, 1 followed by 100 zeros). This number was introduced in 1920 per Milton Sirotta, 9 years old, who was the mathematic American Edward Kasner nephew. This number was used as an example in his book  Mathematics and the Imagination and even though it has not any usefulness in mathematics world, it is used to illustrate the difference between an unimaginable large number and the infinite.

A googol could be written in a conventional way as:
1 googol = 10100   
= 10,000,000,000,000, 000,000,000,000,000,000,000,000,000,
000, 000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000, 000,000,000,000
Related to this number there is googolplex, equivalent to 101 googol = 1010100 

It also exist the googolplexian, equivalent to
1 googolplexian = 101 googolplex
= 101010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,