Chess Masters.

firstly, there is 8x8 tinniest squares area of 1x1. now,the actual part begins. you need to visualize the fact. if you consider squares of area of 2x2, then notice that such squaes are 7 in rows and 7 in columns. that means such squres are 7x7=49 in number. in this way, 8x8+7x7+6x6+5x5+4x4+3x3+2x2+1x1= 204 :)

Consider a chessboard 8*8. Total number of squares are : 8*8+7*7+6*6+5*5+4*4+3*3+2*2+1*1 = 204

For a $n \times n$ board, the total number of squares is

$1^{2}$ + $2^{2}$ + $3^{2}$ + ........ + $n^{2}$

Since, a chessboard is a $8 \times 8$ board, the total number of squares is

$1^{2}$ + $2^{2}$ + $3^{2}$ + ........ + $8^{2}$ = $\boxed{204}$

There is a formula for the sum of squares of the integers which is:

1^2 + 2^2 + 3^2 + ... + n^2

The chessboard is 8 by 8. Therefore,

1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2=204

There are actually 204 squares on a chessboard. Surprised! Here is the explanation. There are 64 (1x1) squares. There are 49 (2x2) squares. There are 36 (3x3) squares. There are 25 (4x4) squares. There are 16 (5x5) squares. There are 9 (6x6) squares. Then there are 4 (7x7) squares and 1 big 8x8 square. So, there are a total of 204 squares on a normal chessboard!!!.