When counting squares in a grid, remember to count squares of all sizes, including smaller ones and the larger squares formed by combining multiple smaller squares.
The given figure consists of a \(4 \times 4\) grid, where we need to count the total number of squares of various sizes.
Step 1: Count the \( 1 \times 1 \) squares Since the grid is \(4 \times 4\), the number of \(1 \times 1\) squares is: \[ 4 \times 4 = 16. \]
Step 2: Count the \( 2 \times 2 \) squares Each \(2 \times 2\) square fits within a \(3 \times 3\) subgrid, so the count is: \[ 3 \times 3 = 9. \]
Step 3: Count the \( 3 \times 3 \) squares Each \(3 \times 3\) square fits within a \(2 \times 2\) subgrid, so the count is: \[ 2 \times 2 = 4. \]
Step 4: Count the \( 4 \times 4 \) square There is only one \(4 \times 4\) square, which is the entire grid itself: \[ 1. \]
Step 5: Compute the total number of squares \[ 16 \, (1 \times 1) + 9 \, (2 \times 2) + 4 \, (3 \times 3) + 1 \, (4 \times 4) = 30. \] Thus, the total number of squares in the grid is \(\mathbf{30}\).
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