
The total numbers of square on a chessboard is:
A) \[206\]
B) \[205\]
C) \[204\]
D) \[202\]
Answer
484.8k+ views
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:
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 \[ = 204\]
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 \[ = 204\]
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