Question

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

Answer 1

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 $= 204$