Chess Board Concept

                Chess Board Concept

A chess board has 8 rows & 8 columns.
Number of squares in chess board ⇒

Number of square of size 1×1 = 8×8 = 64
Number of square of size 2×2 = 7×7 = 49
Number of square of size 3×3 = 6×6 = 36
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Number of square of size 8×8 = 1×1 = 1
Hence total number of squares in chess board = 1² + 2² + 3² + ………… + 8²
            = 204

Number of Rectangle in chess board ⇒

To calculate number of rectangles we have to select 2 lines from 9 verticle lines. Similarly, another set of 2 lines from 9 horizontal lines.
Hence total number of rectangle = ⁹C₂ × ⁹C₂
                                                            = 1296
Here point to remember is that rectangle includes squares in calculation

Thus in chess board of size n×n ⇒

Example:- Find the number of squares in chess board of size 12×12
Solution:- 

1² + 2² + 3² + 4² + ……….. + 12²
Here n =12
∴ number of squares = 
                                       =            
                                       = 13×50
                                       = 650 Answer

Example:- Find the number of rectangle in  chess board of size 12×12.
Solution:-

Here n = 12 
∴ number of rectangle =    
                                          = 
                                          = 78²
                                          = 6084 Answer