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. 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.

Games and puzzles using checkerboards
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.
 * Arimaa
 * Breakthrough
 * Crossings
 * Mak-yek
 * Makruk
 * Martian Chess

Mathematical description
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,

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

The element $$(m,n)=(0,0)$$ is black and represents the lower left corner of the board.