Total no. of squares in the chess board are

204 squares of different sizes

1*1 squares: 8*8 = 64
2*2 squares: 7*7 = 49
3*3 squares: 6*6 = 36
.
.
.
.
7*7 squares: 2*2 = 4
8*8 squares: = 1

Total number of squares = 8^2 +7^2+......+2^2+1^2
= 64+49+36+25+16+9+4+1
= 204

Consider the lefthand vertical edge of a square of size 1 x 1.
This edge can be in any one of 8 positions. Similarly, the top
edge can occupy any one of 8 positions for a 1 x 1 square.
So the total number of 1 x 1 squares = 8 x 8 = 64.

For a 2 x 2 square the lefthand edge can occupy 7 positions and
the top edge 7 positions, giving 7 x 7 = 49 squares of size 2 x 2.

Continuing in this way we get squares of size 3 x 3, 4 x 4 and so on.
Size Of square Number of squares
--------------- -----------------
1 x 1 8^2 = 64
2 x 2 7^2 = 49
3 x 3 6^2 = 36
4 x 4 5^2 = 25
5 x 5 4^2 = 16
6 x 6 3^2 = 9
7 x 7 2^2 = 4
8 x 8 1^2 = 1
---------------
Total = 204
---------------

1*1=64 Squares
2*2=49 Squares
3*3=36 Squares
4*4=25 Squares
5*5=16 Squares
6*6=9 Squares
7*7=4 Squares
8*8=1Squares
Total= 64+49+36+25+16+9+4+1
= 204 Squares

Total number of squares are 8²+7²+6²+5²+4²+3²+2²+1² = 204

Formula is
1² + 2² + 3² + ....n² = n(n+1)(2n+1)/6
(8*9*17)/6 = 4*3*17 = 204

Size Of square Number of squares

1 x 1 8^2 = 64
2 x 2 7^2 = 49
3 x 3 6^2 = 36
4 x 4 5^2 = 25
5 x 5 4^2 = 16
6 x 6 3^2 = 9
7 x 7 2^2 = 4
8 x 8 1^2 = 1
---------------
Total = 204

(8*9*17)/6=204

size horizontal positions vertical positions positons
1x1 8 8 64
2x2 7 7 49
3x3 6 6 36
4x4 5 5 25
5x5 4 4 16
6x6 3 3 9
7x7 2 2 4
8x8 1 1 1
total 204

1x18 8 64
2x2 7 7 49
3x3 6 6 36
4x4 5 5 25
5x5 4 4 16
6x6 3 3 9
7x7 2 2 4
8x8 1 1 1
-----
204
------

8^2 + 7^2 + 6^2 + .... 1= 204

8*8+7*7+6*6+5*5+4*4+3*3+2*2+1*1
= 64+49+36+25+16+9+4+1 = 204
So Ans. 6) 204

1^2+2^2+...+8^2= n(n+1)(2n+1)/6
so, 204 is the answer.

1^2+2^2+3^2....8^2=n*(n+1)*(2n+1)/6
use the formula
8*9*17/6=204

1, 8x8 square
4, 7x7 squares
9, 6x6 squares
16, 5x5 squares
25, 4x4 squares
36, 3x3 squares
49, 2x2 squares
64, 1x1 squares
Therefore, there are actually 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 squares on a chessboard! (in total 204).

There is a formula for the sum of squares of the integers
1^2 + 2^2 + 3^2 + ... + n^2

n(n+1)(2n+1)
Sum = ------------
6

n=8;
sum= n(n+1)(2n+1) / 6;
ans=204

n=8;
sum= n(n+1)(2n+1) / 6;
72*17/6;
ans=204

sigma (n^2)
= n*(n+1)*(2n+1)/6
Here, n = 8
So, result = 8*9*17/6 = 204

total no.of squares=n(n+1)(2n+1)/6
here n=8;
then,
8(9)(17)/6=1224/6=204;
so the answer is:option 6)204

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

1x1 8 8 64
2x2 7 7 49
3x3 6 6 36
4x4 5 5 25
5x5 4 4 16
6x6 3 3 9
7x7 2 2 4
8x8 1 1 1
total 204

size hr vr
1x1 8 8 64
2x2 7 7 49
3x3 6 6 36
4x4 5 5 25
5x5 4 4 16
6x6 3 3 9
7x7 2 2 4
8x8 1 1 1
total 204

Size Of square Number of squares

1 x 1 8^2 = 64
2 x 2 7^2 = 49
3 x 3 6^2 = 36
4 x 4 5^2 = 25
5 x 5 4^2 = 16
6 x 6 3^2 = 9
7 x 7 2^2 = 4
8 x 8 1^2 = 1
---------------
Total = 204

There are 64, 1x1 squares. 49, 2x2 squares..and so on upto 1 8x8 square..which comes to 64+49+36+25+16+9+4+1=204

Total number of squares in 8 x 8 chess board are :
8*2 + 7*2 + 6*2 + 5*2 + 4*2 + 3*2 + 2*2 + 1*2 = 204

because it is in series like 8*8+7*7+.....1*1

1, 8x8 square
4, 7x7 squares
9, 6x6 squares
16, 5x5 squares
25, 4x4 squares
36, 3x3 squares
49, 2x2 squares
64, 1x1 squares

Therefore, there are actually 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 squares on a chessboard! (in total 204).

1x1 squares are: 8^2 = 64
2x2 squares are: 7^2 = 49
3x3 squares are: 6^2 = 36
.......
...
..
8x8 squares are: 1^2= 1

64+49+36+25+16+9+4+1==204

Put n = 8, apply in this (n*(n+1)*(2n+1))/6

1*1=64
2*2=49
3*3=36
4*4=25
5*5=16
6*6=9
7*7=4
8*8=1
64+49+36+25+16+9+4+1=204

1*1=8^2
2*2=7^2
3*3=6^2
4*4=5^2
5*5=4^2
6*6=3^2
7*7=2^2
8*8=1^1

total=204 squares

The sum of the squares of the first n natural numbers is:
n(n+1)(2n+1)/2*3
ans=204

Size Of square Number of squares
--------------- -----------------
1 x 1 8^2 = 64
2 x 2 7^2 = 49
3 x 3 6^2 = 36
4 x 4 5^2 = 25
5 x 5 4^2 = 16
6 x 6 3^2 = 9
7 x 7 2^2 = 4
8 x 8 1^2 = 1
---------------
Total = 204

1, 8x8 square
4, 7x7 squares
9, 6x6 squares
16, 5x5 squares
25, 4x4 squares
36, 3x3 squares
49, 2x2 squares
64, 1x1 squares
Therefore, there are actually 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 squares on a chessboard! (in total 204).

no. of squres
1x1 64
2x2 49
3x3 36
4x4 25
5x5 16
6x6 9
7x7 4
8x8 1
total 204

1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2
=1+4+9+16+25+36+49+64
=204