Question

The total numbers of square on a chessboard is:
A) \[206\]
B) \[205\]
C) \[204\]
D) \[202\]

Answer 1

Hint: At first, we have to find out how many positions there are that each size of square can be located.
A chess board contains \[1 \times 1\], \[2 \times 2\],\[3 \times 3\], \[4 \times 4\],\[5 \times 5\],\[6 \times 6\],\[7 \times 7\],\[8 \times 8\] square located in different places though can only fit in 1 position vertically and 1 horizontally.
We can find the locations for those squares, then we can find the sum of squares.

Complete step-by-step solution:
We have to find the total number of squares on a chessboard.
At first, we have to find out how many positions there are that each size of square can be located.
For example, a \[1 \times 1\] square can be located in 8 locations horizontally and 8 locations vertically that is in 64 different positions. An \[8 \times 8\] square though can only fit in 1 position vertically and 1 horizontally.
For example, a \[2 \times 2\] square can be located in 7 locations horizontally and 7 locations vertically that is in 49 different positions. An \[7 \times 7\] square though can only fit in 2 positions vertically and 2 horizontally.
So, we can prepare a table such as:

Size	Horizontal position	Vertical position	Positions
\[1 \times 1\]	8	8	64
\[2 \times 2\]	7	7	49
\[3 \times 3\]	6	6	36
\[4 \times 4\]	5	5	25
\[5 \times 5\]	4	4	16
\[6 \times 6\]	3	3	9
\[7 \times 7\]	2	2	4
\[8 \times 8\]	1	1	1
		Total	204

Hence, the total number of squares on a chessboard is 204.

Hence, the correct option is C.

Note: It is clear from the above analysis that the solution in case of \[n \times n\] is the sum of the squares from \[{n^2}\] to \[{1^2}\] that is to say
\[{n^2} + {(n - 1)^2} + {(n - 2)^2} + {(n - 3)^2} + ... + {2^2} + {1^2}\]
For a chessboard, \[n = 8\]
So, the total number of squares is \[{8^2} + {7^2} + {6^2} + {5^2} + ... + {2^2} + {1^2}\]
Solving we get, the total number of squares is \[ = 204\]