# Checkerboard

An empty 8×8 checkerboard

A checkerboard (American English) or chequerboard (British English; see spelling differences) is a board of chequered pattern on which draughts (checkers) is played.[1] Most commonly, it consists of 64 squares (8×8) of alternating dark and light color, often black and white. 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.

## Mathematical Description

An empty 8×8 checkerboard diagram

Given a matrix with ${\displaystyle m}$ rows and ${\displaystyle n}$ columns, a function ${\displaystyle f(m,n)}$,

${\displaystyle \displaystyle {f(m,n)}={\begin{cases}{\text{true}}&{\text{if}}\ m\wedge 1=1\ {\text{and}}\ n\wedge 1=1\,,\\{\text{false}}&{\text{if}}\ m\wedge 1=1\ {\text{and}}\ n\wedge 1=0\,,\\{\text{false}}&{\text{if}}\ m\wedge 1=0\ {\text{and}}\ n\wedge 1=0\,,\\{\text{true}}&{\text{if}}\ m\wedge 1=0\ {\text{and}}\ n\wedge 1=1\,,\\\end{cases}}}$

returns true if the cell is white and false if the cell is black. The element ${\displaystyle (m,n)=(0,0)}$ is black.

## Games and puzzles using checkerboards

The opening setup of international draughts, which uses a 10×10 checkerboard

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:

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