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

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

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

Amazons
Chapayev
Chess and some of its variants (see chessboard)
Czech draughts
Draughts
Frisian draughts
Gounki
International draughts
Italian draughts
Lines of Action
Pool checkers
Russian checkers
Turkish draughts

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

References

↑ http://mathworld.wolfram.com/Checkerboard.html

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[1] ttp://mathworld.wolfram.com/Checkerboard.html