Solution:
All the rectangles on the board can be identified by connecting:
2 points of the 7 in the top edge (to form the length of the rectangle) and
2 points of the 7 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 6 × 6 board:
Length of rectangle | Number of Possibilities |
6 units | 1 |
5 units | 2 |
4 units | 3 |
... | ... |
1 unit | 6 |
So, number of possibilities for different lengths of rectangles = 1 + 2 + 3 + ... + 6 = 21.
Similarly, number of possibilities for different breadths of rectangles = 1 + 2 + 3 + ... + 6 = 21.
Hence, number of rectangles = 21 × 21 = 441.
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
nC
2 is the number of combinations of
n things taken 2 at a time.
nC
2 =
n (
n − 1)/2. Hence the number of rectangles =
7C
2 ×
7C
2 = 21 × 21 = 441.
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.