Question

Number of squares in a chess-board is equal to:

• A. 64
• B. 81
• C. 204
• D. 284

Hint:

To find the total number of squares in a grid, sum the number of squares of each possible size. For an n x n grid, you sum the number of 1x1, 2x2, ..., nxn squares.

Answer 1

The Correct Option is C

A standard chessboard is an 8x8 grid, but we need to find the total number of squares of all sizes that can be formed on the chessboard.
Step 1: Count of 1x1 squares.
The number of 1x1 squares is simply the number of small squares on the board: 8 × 8 = 64 1x1 squares.
Step 2: Count of 2x2 squares.
To form a 2x2 square, we can choose the top-left corner from any of the first 7 rows and 7 columns (since a 2x2 square takes up two rows and two columns). Thus, there are: 7 × 7 = 49 2x2 squares.
Step 3: Count of 3x3 squares.
Similarly, for 3x3 squares, we can choose the top-left corner from any of the first 6 rows and 6 columns. Thus, there are: 6 × 6 = 36 3x3 squares.
Step 4: Count of 4x4 squares.
For 4x4 squares, we can choose the top-left corner from any of the first 5 rows and 5 columns. Thus, there are: 5 × 5 = 25 4x4 squares.
Step 5: Count of 5x5 squares.
For 5x5 squares, we can choose the top-left corner from any of the first 4 rows and 4 columns. Thus, there are: 4 × 4 = 16 5x5 squares.
Step 6: Count of 6x6 squares.
For 6x6 squares, we can choose the top-left corner from any of the first 3 rows and 3 columns. Thus, there are: 3 × 3 = 9 6x6 squares.
Step 7: Count of 7x7 squares.
For 7x7 squares, we can choose the top-left corner from any of the first 2 rows and 2 columns. Thus, there are: 2 × 2 = 4 7x7 squares.
Step 8: Count of 8x8 squares.
Finally, there is only 1 square of size 8x8, which is the entire chessboard itself: 1 × 1 = 1 8x8 square.
Step 9: Total number of squares.
The total number of squares is the sum of all the squares of different sizes: 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 total squares.
Therefore, the correct answer is (3) 204.