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Rectangles on Checkered Board

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1. If you were to construct a 5 × 5 checkered square (i.e., a 5 × 5 chess board), how many rectangles would there be in total? You need to include squares too because a square is a special kind of rectangle.
Answer: 225

Solution:

All the rectangles on the board can be identified by connecting:
2 points of the 6 in the top edge (to form the length of the rectangle) and
2 points of the 6 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 5 × 5 board:

Length of rectangle Number of Possibilities
5 units 1
4 units 2
3 units 3
... ...
1 unit 5

So, number of possibilities for different lengths of rectangles = 1 + 2 + 3 + ... + 5 = 15.
Similarly, number of possibilities for different breadths of rectangles = 1 + 2 + 3 + ... + 5 = 15.
Hence, number of rectangles = 15 × 15 = 225.

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 nC2 is the number of combinations of n things taken 2 at a time.
nC2 = n (n − 1)/2. Hence the number of rectangles = 6C2 × 6C2 = 15 × 15 = 225.

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.

  Try the Quiz :     Puzzles & Brain Teasers : Rectangles on Checkered Board

puzzle : image for chess board

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