The figure shows a grid formed by a collection of unit squares. The unshaded unit square in the grid represents a hole. What is the maximum number of squares without a "hole in the interior" that can be formed within the 4 $\times$ 4 grid using the unit squares as building blocks?
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To maximize the number of squares without a "hole in the interior," it is important to consider the sizes of squares and avoid placing the hole within the boundaries of any square.
Step 1: Understanding the structure of the grid
The grid has a total of 16 unit squares. One of these unit squares is a hole in the center. Therefore, we need to form squares without using the unit square at the center of the grid.
Step 2: Finding possible square sizes
- A $1 \times 1$ square can be formed in any of the 15 remaining unit squares (excluding the center hole).
- A $2 \times 2$ square can be formed by selecting four unit squares. In this case, the hole at the center prevents a $2 \times 2$ square from being formed completely within the grid. Thus, we can form 5 such $2 \times 2$ squares.
- A $3 \times 3$ square can be formed by selecting a $3 \times 3$ block of squares. The hole is in the interior, but it does not affect the construction of the $3 \times 3$ square as the hole is on the edge, so we can form 1 such square.
Step 3: Summing the possible squares
Total number of squares that can be formed:
- 15 squares of size $1 \times 1$
- 5 squares of size $2 \times 2$
- 1 square of size $3 \times 3$
Thus, the maximum number of squares that can be formed without a "hole in the interior" is:
\[
15 + 5 + 1 = 20
\]
