How many squares are there on a chessboard? (Hint: The answer is not 64)

Answer

204 squares.

The chessboard is made up of 8×8 squares. We have to determine how many squares of any dimension from 1×1 to 8×8 there are on a chess board.

The key is to determine how many possible positions there are for each type of square. A 2×2 square, for example, can be located in 7 locations horizontally and 7 locations vertically. eg. in 49 different positions.

size | horizontal positions | vertical positions | squares | |
---|---|---|---|---|

1x1 | 8 | 8 | 64 | |

2x2 | 7 | 7 | 49 | |

3x3 | 6 | 6 | 36 | |

4x4 | 5 | 5 | 25 | |

5x5 | 4 | 4 | 16 | |

6x6 | 3 | 3 | 9 | |

7x7 | 2 | 2 | 4 | |

8x8 | 1 | 1 | 1 | |

Total | 204 |

In total there are 204 positions. this is the sum of the number of possible positions for all the different sized squares.