How many Chess games are possible?

Chess, as we know it today, is believed to be originated from Chaturanga, an Indian game which was played before 600 AD. The game spread throughout Asia and Europe over the coming centuries, and eventually evolved into what we know as chess today around the 16th century.

You may be knowing different tricks to win a chess game, but do you know “How many chess games are possible?"

The number of possible chess games is 10^120, also known as Shannon’s number and this is just an estimate. Now what is Shannon’s number and how does it work out? An American scientist named Claude Shannon, in 1950s wrote a paper on how to program a computer to play chess and in that paper he arrived to an estimate of how many games of chess is possible to this number. Now the number of the atoms in the observable universe is around 10^80 and the number of stars are about 10^24. So there are BILLIONS of games for each atom in the universe.

Now, how did Shannon arrive to this number? So, he observed a few games of chess and came to a conclusion that on average after every position there are about 30 legal moves that can be played. So on average first player has 30 legal moves for each move of the first player, the second player also has 30 legal moves. So, there are (30 x 30=) 900 possible games just in 1 move. In chess, when both the players move a piece that is considered as 1 move.

So, if you consider the starting position on the board. There are 20 ways in which white can start the game.

By moving a pawn just 1 step, which makes 8 possible games.

Or by moving a pawn by 2 steps, which again makes 8 possible games,

Or by moving either of the knight to any of the 2 squares available for each knight.

Which makes a total of 20 possible plies, remember this is just a half move, the response from the black pieces is yet to be consider, which is the exact same move as white’s for each move from the white.

Now, here we have seen just 20 plies then how did Shannon consider 30 plies. It was the average of all the possible half moves possible at all the any position of the board.

Let’s look at an example,

Here in this very complicated picture of a chess board, there are 42 possible plies which can be made by white. Each pawn can be moved in 2 ways so that’s 10 plies, Bishops have 9 possible squares to move, knights have 9 squares, Queen can move to any of the 4 squares available, rooks have 5 squares and king can move in any of the 3 squares around it, that makes 40 plies what about the remaining 2. Both long and short castling is available for the white so that makes 42 possible plies.

So, on average there will be 30 plies possible for each player.

Now, taking this into consideration if we calculate the total number of games just by considering a single move played, we will get 900 different games.

Now, Shannon considered only the number of “sensible” games that can be played, not counting ridiculous or obvious game-losing moves such as moving a queen to be immediately captured by a pawn without compensation, then the result is about 40 moves (by each player). So, total of 80 plies will give us 30^80 = 10^120.

Remember this is just an estimation and the game is considered to be of just 40 moves. But a longest chess game can be something around 5900 moves, that’s 11800 plies. And this is considering the 50 move rule. This rule says that if you move 50 moves without nothing being captured then the game will result in a draw.

Why should we take the average of all the possible moves which can be made, let’s see what will happen if we start from 20 ply and take the number of moves according to each move made,

And the list goes on. In these 5900 moves some of the moves will definitely be non-sense moves. Like even if you can win in one move, you start to do something else, I mean how you would get the longest game without that. Now, considering the increase rate in just 5 moves, just imagine the number at end of 5900th move, it will be this massive number which to find seems impossible.

Godfrey Hardy, a famous 20th century Mathematician tried to estimate the number of chess games. His estimation was 10^(10^50), Ten to the power of ten followed by 50 zeros, compared to this 10^120 is very, very, very small number. With this comes another arrgument: is this number even close to the correct answer to the number to possible games of chess? Nobody knows! As Hardy didn’t give a proof of this, he just gave some number that he thought was a possible answer.

So, if considering only the sensible moves instead of all the legal moves available then Shannon’s estimation seems right.