# Chessboard squares and rectangles

Another problem I posed a month or so ago:

“*How many (a) squares and (b) rectangles (with whole number side lengths) are there on a chessboard?*”

Again, I’ll give you some thinking space…

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

** **

Many, many problems in maths are, in the end, about counting things, and so the problem is actually about finding a good way to count. These are no exception.

Naively, we could start to count every different square (or rectangle) we see. For small numbers that might be OK, but here it would be quite impossible – at least for me. I’d be sure to forget to count one, or not notice that I had counted one twice.

**(a) Squares**

As usual, there would be many ways to do this. My suggestion is …

For each square on the chessboard, count the squares (of all sizes) having that square in their top-left-hand corner, as follows:

8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |

7 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |

6 | 6 | 6 | 5 | 4 | 3 | 2 | 1 |

5 | 5 | 5 | 5 | 4 | 3 | 2 | 1 |

4 | 4 | 4 | 4 | 4 | 3 | 2 | 1 |

3 | 3 | 3 | 3 | 3 | 3 | 2 | 1 |

2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

So the total of all the numbers in this grid is

1×8 + 3×7 + 5×6 + 7×5 + 9×4 + 11×3 +13×2 + 15×1 = 204.

More generally, for nxn chessboards, the number of squares is n(n+1)(2n+1)/6.

**(b) Rectangles**

The equivalent counts for rectangles on the chessboard are:

64 | 56 | 48 | 40 | 32 | 24 | 16 | 8 |

56 | 49 | 42 | 35 | 28 | 21 | 14 | 7 |

48 | 42 | 36 | 30 | 24 | 18 | 12 | 6 |

40 | 35 | 30 | 25 | 20 | 15 | 10 | 5 |

32 | 28 | 24 | 20 | 16 | 12 | 8 | 4 |

24 | 21 | 18 | 15 | 12 | 9 | 6 | 3 |

16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 |

8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |

The bottom row is just the sum of the numbers 1 to 8 = 36.

The 2nd bottom row is just 2 times the sum of the numbers 1 to 8 = 2 x 36.

The 3rd bottom row is just 3 times the sum of the numbers 1 to 8 = 3 x 36.

The 4th bottom row is just 4 times the sum of the numbers 1 to 8 = 4 x 36.

The 5th bottom row is just 5 times the sum of the numbers 1 to 8 = 5 x 36.

The 6th bottom row is just 6 times the sum of the numbers 1 to 8 = 6 x 36.

The 7th bottom row is just 7 times the sum of the numbers 1 to 8 = 7 x 36.

The 8th bottom (i.e. top) row is 8 times the sum of the numbers 1 to 8 = 8 x 36.

So the total of all the numbers in this grid is

(1+2+3+4+5+6+7+8) x 36 = 36 x 36 = 1296.

Check it yourself.

More generally, for nxn chessboards, the number of squares is n^{2}(n+1)^{2}/4.