# Checkerboard

5 by 5 checkerboard pattern
Checkered pattern on paramedic fly-car
Checkered pattern on floor in a bathroom

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

Other rectangular square-tiled boards are also often called checkerboards. In this sense it refers not to a physical board as such but to the mathematical abstraction of such a board. The adjective chequered refers to the pattern shown in many contexts, such as the checkered flag used to signify the end of a vehicle race, the popular ice cream flavor or the livery on some emergency service vehicles. However, when this pattern is used on such vehicles, in certain countries, it is called a Sillitoe Tartan. A red-and-white checkerboard is the medieval coat of arms of Croatia, and is widely used in national symbology there.

The checkerboard pattern is often associated with the Ska music genre, for breaking the racial barrier between black and whites at the time. A checkerboard marking painted on a hilltop was used as visual guidance on Hong Kong's old Kai Tak Airport. Many taxicabs also use a checkerboard pattern. The comic book publishing company, DC Comics, briefly tried marking their publications in the 1960s with an upper border checkerboard pattern as a marketing strategy to make them stand out on newsstands.[2]

A checkerboard pattern of alternating aviation orange and white is commonly required for compliance with the FAA as a means of making tall or sufficiently sizable structures stand out, so as to make them clearly visible to pilots, including water tanks, gas tanks, grain storage tanks, tall buildings and flags.[3]

## Mathematical Description

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

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.