Question

How many squares are there in the given figure?

• A. \(32\)
• B. \(48\)
• C. \(78\)
• D. \(70\)

Hint:

In grid problems, never stop at counting unit squares—always look for larger squares formed by combining them.

Answer 1

The Correct Option is D

To determine the total number of squares in the given figure, we need to consider squares of different sizes that can be formed in the grid shown.

The figure is a grid with 4 rows and 8 columns. We will calculate the number of squares of size \(1 \times 1\), \(2 \times 2\), \(3 \times 3\), and \(4 \times 4\).

1. Squares of size \(1 \times 1\):
   • Each cell in the grid is a square. There are \(4\) rows and \(8\) columns, resulting in \(4 \times 8 = 32\) squares of size \(1 \times 1\).
2. Squares of size \(2 \times 2\):
   • Each \(2 \times 2\) square covers four adjacent cells. The number of such squares is determined by the number of possible positions for the top-left corner of each square.
   • There are \(3\) possible positions for the top-left corner in each row and \(7\) possible positions in each column, resulting in \(3 \times 7 = 21\) squares of size \(2 \times 2\).
3. Squares of size \(3 \times 3\):
   • Each \(3 \times 3\) square covers nine cells. The count is determined by possible positions for the top-left corner.
   • There are \(2\) possible positions for the top-left corner in each row and \(6\) possible positions in each column, resulting in \(2 \times 6 = 12\) squares of size \(3 \times 3\).
4. Squares of size \(4 \times 4\):
   • Each \(4 \times 4\) square covers sixteen cells, with the top-left corner able to be in certain positions.
   • There is \(1\) possible position for the top-left corner in each row and \(5\) possible positions in each column, resulting in \(1 \times 5 = 5\) squares of size \(4 \times 4\).

Now, add the number of squares of all sizes:

\[ 32 \, (1 \times 1) + 21 \, (2 \times 2) + 12 \, (3 \times 3) + 5 \, (4 \times 4) = 70 \]

Therefore, the total number of squares in the grid is 70.