Why are there 64 squares on a chessboard? My question is, why not 36, 49, 81 or some other square number? Do any historical sources point to how and why chess came to be played on 64 squares in particular? Has it always been so?
Nothing is stopping you from playing chess in a 4x4, 6x6 or 9x9 board. In ancient times people have tried such approaches.

To answer why 64 squares, I have to answer a bit mathematically. Let me start with this:


  [Chess, in] its early form in the 6th century was known as chaturaṅga, which translates as "four divisions (of the military)": infantry, cavalry, elephantry, and chariotry.


It says that chaturanga means 'game of squares' and it also mentions 4 divisions of military, where 1 division = 8 pieces (4 pawns + 4 main pieces). So 4x4 = 16 pieces each side. This also means, all in all, a total of 32 pieces in the board (8 in each row).

For 32 pieces to be fully mobile on the board, 36 squares would be too congested and not possible; 49 squares would be still too congested; 64 sure makes sense, and is also the perfect square of 8.