A recent episode of Numberphile gives a mathematical explanation of the number of possible chess games. In the episode, Dr. James Grime calls into question the Shannon number of 10120, the commonly accepted number of chess games, but one that was reached without much careful thought.
Grime takes a more measured approach to explain both the total possible games based on the longest possible chess game, but also calculates the arguably more important distinction of the number of possible sensible chess games where both players are making thoughtful moves, and arrives at the number 1040.
