**Solution:**
All the rectangles on the board can be identified by connecting:

2 points of the 9 in the top edge (to form the length of the rectangle) and

2 points of the 9 in the left edge (to form the breadth of the rectangle).

To gain a better understanding, consider the 8 × 8 chess board (see animation above).

Note that there are 4 possibilities for the lengths of the rectangles to be 5 units.

The following table shows the number of possibilities for different lengths of the rectangles on a 8 × 8 board:

Length of rectangle | Number of Possibilities |

8 units | 1 |

7 units | 2 |

6 units | 3 |

... | ... |

1 unit | 8 |

So, number of possibilities for different lengths of rectangles = 1 + 2 + 3 + ... + 8 = 36.

Similarly, number of possibilities for different breadths of rectangles = 1 + 2 + 3 + ... + 8 = 36.

Hence, number of rectangles = 36 × 36 = 1296.

**Food for thought:**
Is there a formula for the sum of the first

*n* positive integers ?

Is 1 + 2 + 3 + 4 + ... +

*n* =

*n* (

*n* + 1) / 2 ?

Can this puzzle be solved quickly with knowledge of permutations and combinations?

Note

^{n}C

_{2} is the number of combinations of

*n* things taken 2 at a time.

^{n}C

_{2} =

*n* (

*n* − 1)/2. Hence the number of rectangles =

^{9}C

_{2} ×

^{9}C

_{2} = 36 × 36 = 1296.

Can you figure out the following alternative formula to solve this puzzle?

Number of rectangles on

*n* ×

*n* board

= 2 (Sum of the products of all pairs of numbers from 1 to

*n*) − (Number of squares on the board)

So, how many squares on a

*n* ×

*n* board?

Click here to find out.