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 $$=204$$