Number of positions in chess after n moves

Chess mathematics can be fascinating. At first sight chess seems to be easy to calculate. It has logical patterns and finite board space. However, the simplest questions may require serious mathematical skills.

A good example is the number of possible positions after n moves, n being 1, 2, 3, etc. After the first move there are exactly 20 positions, after the second, there are 400. White has a choice of 20 first moves, Black the same number of replies, making 400 different possible positions after one move for each color. From here on it is difficult to keep on counting since the number is rapidly growing. After the third move we have 5362 positions, and after the fourth the number is 71852. A really large number for 8×8 board!

These numbers are a good back up of the complexity of The number of Shannon. In 1889 Cunningham got close to the number of moves after the 4th move, stating they are 71782. Fabel got even closer in 1895, he calculated 71870 possible moves. The first one to find the correct number, 71852, was C. Flye St. Marie in 1903.

As far as the Chessdom team knows, there are estimations of the number of positions after the 5th and the 6th moves. They are 809798 and 9132484 respectively. However, we would like to receive confirmation or a more correct information from our mathematician readers. Do not forget to include special moves like en passant .