Chess960

Currently an unofficial world championship is underway in Oslo pitting Hikaru Nakamura against Magnus Carlsen. However, the two are not playing traditional chess but instead so-called Chess960 described as follows in Wikipedia:

Chess960, also called Fischer Random Chess (originally Fischerandom), is a variant of chess invented and advocated by former world chess champion Bobby Fischer, publicly announced on June 19, 1996, in Buenos Aires, Argentina. 

It employs the same board and pieces as standard chess, but the starting position of the pieces on the players' home ranks is randomised. The random setup renders the prospect of obtaining an advantage through the memorisation of opening lines impractical, compelling players to rely on their talent and creativity. 

Randomising the main pieces had long been known as Shuffle Chess; however, Chess960 introduces restrictions on the randomisation, "preserving the dynamic nature of the game by retaining bishops of opposite colours for each player and the right to castle for both sides".[3] The result is 960 unique possible starting positions. 

White's pieces (not pawns) are placed randomly on the first rank, with two restrictions:

• The bishops must be placed on opposite-color squares.

• The king must be placed on a square between the rooks.

• Black's pieces are placed equal-and-opposite to White's pieces. (For example, if the white king is randomly determined to start on f1, then the black king is placed on f8.) Pawns are placed on the players' second ranks as in standard chess.

It is the number 960 that is of interest in this mathematics blog and this is explained as follow:

Each bishop can take one of four squares; for each position of two bishops, the queen can be placed on six different squares; and then the two knights can assume five and four possible squares, respectively. This leaves three open squares which the king and rooks must occupy, per setup stipulations, without choice. This means there are 4×4×6×5×4 = 1920 possible starting positions if the two knights were different in some way; however, the two knights are indistinguishable during play (if swapped, there would be no difference), so the number of distinguishable possible positions is half of 1920, or 1920÷2 = 960. (Half of the 960 positions are left–right mirror images of the other half; however, the Chess960 castling rules preserve left–right asymmetry in play.) 

There is another variant called Double Fischer Random Chess which is the same as Chess960, except the White and Black starting positions do not mirror each other. In this form of the game, the number of possible starting positions is 960 x 960 = 921600. In Shuffle Chess, the parent variant of Chess960, there are no restrictions on the back-rank shuffles, with castling possible only when king and rook are on their traditional starting squares. In the case, the number of permutations is 8! but division by 8 is necessary because the pairs of Knights, Bishops and Rooks are indistinguishable. This means 7! or 5040 arrangements are possible.