Checkerboard

A checkerboard (American English) or chequerboard (British English; see spelling differences) is a game board of checkered pattern on which checkers (also known as English draughts) is played.^[1] Most commonly, it consists of 64 squares (8×8) of alternating dark and light color, typically green and buff (official tournaments), black and red (consumer commercial), or black and white (printed diagrams). An 8×8 checkerboard is used to play many other games, including chess, whereby it is known as a chessboard. Other rectangular square-tiled boards are also often called checkerboards. In The Netherlands, however, a dammenbord (checker board) has 10 rows and 10 columns for 100 squares in total (see article International draughts).

Martin Gardner featured puzzles based on checkerboards in his November 1962 Mathematical Games column in Scientific American. A square checkerboard with an alternating pattern is used for games including:

Amazons
Chapayev
Chess and some of its variants (see chessboard)
Czech draughts
Draughts, also known as checkers
Fox games
Frisian draughts
Gounki
International draughts
Italian draughts
Lines of Action
Pool checkers
Russian checkers

The following games require an 8×8 board and are sometimes played on a chessboard.

Given a grid with $m$ rows and $n$ columns, a function $f(m,n)$ ,

$\displaystyle {f(m,n)}={\begin{cases}{\text{black}}&{\text{if}}\ m\equiv n{\pmod {2}}\,,\\{\text{white}}&{\text{if}}\ m\not \equiv n{\pmod {2}}\\\end{cases}}$

or, alternatively,

$\displaystyle {f(m,n)}={\begin{cases}{\text{black}}&{\text{if}}\ m+n{\text{ is even}},\\{\text{white}}&{\text{if}}\ m+n{\text{ is odd}}\\\end{cases}}$