Question

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$calculate \: no.\: of \: square \: in \: chessboard$
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Answer 1

Method-1

We know in a chessboard,

• it contains 1×1, 2×2, 3×3, 4×4, 5×5, 6×6, 7×7, 8×8 square located in different places though can only fit in 1 position vertically and 1 horizontally.

So,

• For a 1×1 square, it can be located 8 places horizontally and 8 places vertically. ⇒ 8×8 = 64 squares.

• For a 2×2 square, it can be located 7 places horizontally and 7 places vertically. ⇒ 7×7 = 49 squares.

• For a 3×3 square, it can be located 6 places horizontally and 6 places vertically. ⇒ 6×6 = 36 squares.

• For a 4×4 square, it can be located 5 places horizontally and 5 places vertically. ⇒ 5×5 = 25 squares.

• For a 5×5 square, it can be located 4 places horizontally and 4 places vertically. ⇒ 4×4 = 16 squares.

• For a 6×6 square, it can be located 3 places horizontally and 3 places vertically. ⇒ 3×3 = 9 squares.

• For a 7×7 square, it can be located 2 places horizontally and 2 places vertically. ⇒ 2×2 = 4 squares.

• For a 8×8 square, it can be located 1 places horizontally and 1 places vertically. ⇒ 1×1 = 1 squares.

Adding these, we get = 64+49+36+25+16+9+4+1 = 204 squares.

Method-2

Now, it is clear that the solution in case of n×n  is the sum of the squares from n² to 1².

$\boxed{\bold{\sf{ Sum \ of \ Squares=n^2+(n-1)^2+(n-2)^2+...+2^2+1^2}}}$

As we know in case of chessboard, n=8.

$\sf{\implies Sum \ of \ Squares=8^2+7^2+6^2+5^2+4^2+3^2+2^2+1^2}$

$\sf{\implies Sum \ of \ Squares=64+49+36+25+16+9+4+1}$

$\sf{\implies Sum \ of \ Squares=204}$

$\huge{\boxed{\bold{\sf{\therefore Sum \ of \ Squares=204}}}}$

Answer 2

Answer:

There are 64 squares in chessboard

Step-by-step explanation:

There are 64 squares in chessboard